optimal continuous dependence estimates for fractional degenerate parabolic equations we derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion the diffusion term involves the fractional laplace operator triangle alpha 2 for alpha in 0 2 our results are quantitative and we exhibit an example for which they are optimal we cover the dependence on the nonlinearities and for the first time the lipschitz dependence on in the bv framework the former estimate dependence on nonlinearity is robust in the sense that it is stable in the limits alpha downarrow 0 and alpha uparrow 2 in the limit alpha uparrow 2 triangle alpha 2 converges to the usual laplacian and we show rigorously that we recover the optimal continuous dependence result of cockburn and gripenberg j differ equ 151 2 231 251 1999 for local degenerate parabolic equations thus providing an alternative proof

the evolution fractional p laplacian equation in rn fundamental solution and asymptotic behaviour abstract we consider the natural time dependent fractional p laplacian equation posed in the whole euclidean space with parameters p 2 and s 0 1 fractional exponent we show that the cauchy problem for data in the lebesgue l q spaces is well posed and show that the solutions form a family of non expansive semigroups with regularity and other interesting properties as main results we construct the self similar fundamental solution for every mass value m and prove that general finite mass solutions converge towards that fundamental solution having the same mass and convergence holds in all l q spaces a number of additional properties and estimates complete the picture

superposition principle for non local fokker planck kolmogorov operators we prove the superposition principle for probability measure valued solutions to non local fokker planck kolmogorov equations which in turn yields the equivalence between martingale problems for stochastic differential equations with jumps and such non local partial differential equations with rough coefficients as an application we obtain a probabilistic representation for weak solutions of fractional porous media equations

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

fractional nonlinear degenerate diffusion equations on bounded domains part i existence uniqueness and upper bounds abstract we investigate quantitative properties of nonnegative solutions u t x 0 to the nonlinear fractional diffusion equation t u l f u 0 posed in a bounded domain x r n with appropriate homogeneous dirichlet boundary conditions as l we can use a quite general class of linear operators that includes the two most common versions of the fractional laplacian s 0 s 1 in a bounded domain with zero dirichlet boundary conditions but it also includes many other examples since our theory only needs some basic properties that are typical of linear heat semigroups the nonlinearity f is assumed to be increasing and is allowed to be degenerate the prototype is the power case f u u m 1 u with m 1 in this paper we propose a suitable class of solutions of the equation and cover the basic theory we prove existence uniqueness of such solutions and we establish upper bounds of two forms absolute bounds and smoothing effects as well as weighted l 1 estimates the class of solutions is very well suited for that work the standard laplacian case s 1 is included and the linear case m 1 can be recovered in the limit in a companion paper bonforte and vazquez in preparation we will complete the study with more advanced estimates like the upper and lower boundary behaviour and harnack inequalities for which the results of this paper are needed

stable and finite morse index solutions of toda system abstract we prove a liouville type theorem and a partial regularity result for stable and finite morse index solutions of toda system the main tools are integral estimates a monotonicity formula and blowing down analysis for such systems they may be split into two sub systems during the blowing down process some e regularity theorems are developed to deal with this problem

nonlocal filtration equations with rough kernels abstract we study the nonlinear and nonlocal cauchy problem t u l u 0 in r n r u 0 u 0 where l is a levy type nonlocal operator with a kernel having a singularity at the origin as that of the fractional laplacian the nonlinearity is nondecreasing and continuous and the initial datum u 0 is assumed to be in l 1 r n we prove existence and uniqueness of weak solutions for a wide class of nonlinearities including the porous media case u u m 1 u m 1 these solutions turn out to be bounded and holder continuous for t 0 we also describe the large time behaviour when the nonlinearity resembles a power for u 0 and the kernel associated to l is close at infinity to that of the fractional laplacian

partial regularity of stable solutions to the supercritical equations and its applications abstract we prove the partial regularity of stable solutions of supercritical elliptic equations as an application we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric

finite element simulation and efficient algorithm for fractional cahn hilliard equation abstract this article is devoted to designing efficient linear finite element algorithm for the fractional cahn hilliard equation an important and newly proposed phase field model combining the advantages of the classic bicg algorithm and the toeplitz like structure of the coefficient matrix we develop a fast bicg fbicg algorithm for the linearized scheme to compute numerically the fractional cahn hilliard equation our theoretical analysis and numerical experiments demonstrate that the proposed fbicg reduces the computation cost and the storage to o m log m and o m possesses the same convergence rates as newton s algorithm does in space and time and preserves the energy dissipation and equality laws the numerical experiments also demonstrate that the fbicg is almost mass conserved recognizes accurately the phase separation by a very clear coarse graining process and the influences of different indices r and s of fractional laplacian and different coefficients k and a on the width of the interfaces

functional inequalities on arbitrary riemannian manifolds abstract for any connected not necessarily complete riemannian manifold we construct a probability measure of type e v x d x where d x is the riemannian volume measure and v is a function c smooth outside a closed set of zero volume satisfying poincare sobolev type functional inequalities in particular v is c smooth on the whole manifold when the poincare and the super poincare inequalities are considered the sobolev inequality for infinite measures are also studied

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

hypocoercivity in metastable settings and kinetic simulated annealing combining classical arguments for the analysis of the simulated annealing algorithm with the more recent hypocoercive method of distorted entropy we prove the convergence for large time of the kinetic langevin annealing with logarithmic cooling schedule

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

a pde approach to fractional diffusion a space fractional wave equation we study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers of order s in 0 1 of symmetric coercive linear elliptic second order operators in bounded domains varomega we realize fractional diffusion as the dirichlet to neumann map for a nonuniformly elliptic problem posed on the semi infinite cylinder mathcal c varomega times 0 infty we thus rewrite our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition and derive space time and space time regularity estimates for its solution the latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation we propose and analyze two fully discrete schemes the discretization in time is based on finite difference discretization techniques the trapezoidal and leapfrog schemes the discretization in space relies on the tensorization of a first degree fem in varomega with a suitable hp fem in the extended variable for both schemes we derive stability and error estimates we consider a first degree fem in varomega with mesh refinement near corners and the aforementioned hp fem in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open bounded polytopal but not necessarily convex domains varomega subset mathbb r 2 we discuss implementation details and report several numerical examples

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

qualitative behavior and traveling waves for flux saturated porous media equations arising in optimal mass transportation abstract this paper is devoted to the analysis of qualitative properties of porous media flux saturated type operators which combine the effects of reaction terms of fkpp type in dimension one specifically we study the existence of traveling wave profiles sub and super solutions waiting time and dynamical properties of the support the paper also deals with the connection between these minimal traveling waves with those of other related models porous media flux saturated mechanism and the heat equation

the toda system and clustering interfaces in the allen cahn equation we consider the allen cahn equation varepsilon 2 delta u 1 u 2 u 0 in a bounded smooth domain in mathbb r 2 under zero neumann boundary conditions where varepsilon 0 is a small parameter let 0 be a segment contained in connecting orthogonally the boundary under certain nondegeneracy and nonminimality assumptions for 0 satisfied for instance by the short axis in an ellipse we construct for any given n 1 a solution exhibiting n transition layers whose mutual distances are o varepsilon log varepsilon and which collapse onto 0 as varepsilon to 0 asymptotic location of these interfaces is governed by a toda type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary

isoperimetric inequalities and markov chains soit a i j i j n la matrice stochastique d une chaine de markov reversible sur n 1 2 soit n 2 tel que f r c f d f c 0 n ou r 2n n 2 ou c 0 est independant de f et ou r indique la norme dans l r n d pour la mesure i i alors on a sup i j j 1 a i j 0 n 2

sharp short and long time l bounds for solutions to porous media equations with homogeneous neumann boundary conditions abstract we study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains r n n 3 with homogeneous neumann boundary conditions we improve the known results in such model case proving sharp l q 0 l regularizing properties of the evolution for short time and sharp long time l bounds for convergence of solutions to their mean value the generality of the discussion allows to consider almost at the same time weighted versions of the above equation provided an appropriate weighted sobolev inequality holds in fact we show that the validity of such weighted sobolev inequality is equivalent to the validity of a suitable l q 0 l bound for solutions to the associated weighted porous media equation the long time asymptotic analysis relies on the assumed weighted sobolev inequality only and allows to prove uniform convergence to the mean value with the rate predicted by linearization in such generality

well posedness of a fractional porous medium equation on an evolving surface abstract we investigate the existence uniqueness and l 1 contractivity of weak solutions to a porous medium equation with fractional diffusion on an evolving hypersurface to settle the existence we reformulate the equation as a local problem on a semi infinite cylinder regularise the porous medium nonlinearity and truncate the cylinder then we pass to the limit first in the truncation parameter and then in the nonlinearity and the identification of limits is done using the theory of subdifferentials of convex functionals in order to facilitate all of this we begin by studying in the setting of closed riemannian manifolds and sobolev spaces the fractional laplace beltrami operator which can be seen as the dirichlet to neumann map of a harmonic extension problem a truncated harmonic extension problem will also be examined and convergence results to the solution of the harmonic extension will be given for a technical reason we will also consider some related extension problems on evolving hypersurfaces which will provide us with the minimal time regularity required on the harmonic extensions in order to properly formulate the moving domain problem this functional analytic theory is of course independent of the fractional porous medium equation and will be of use generally in the analysis of fractional elliptic and parabolic problems on manifolds

further properties of a continuum of model equations with globally defined flux to develop an understanding of singularity formation in vortex sheets we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze a model equation is obtained by replacing the flux term in burgers equation by alternatives that contain contributions depending globally on the solution we consider the continuum of partial differential equationsut h u u x 1 u ux uxx 0 1 0 whereh u is the hilbert transform ofu we show that when 1 2 for 0 the solution of the equation exists for all time and is unique we also show with a combination of analytical and numerical means that the solution when 1 2 and 0 is analytic using a pseudo spectral method in space and the adams moulton fourth order predictor corrector in time we compute the numerical solution of the equation with 1 2 for various viscosities the results confirm that for 0 the solution is well behaved and analytic the numerical results also confirm that for 0 and 1 2 the solution becomes singular in finite time and finite viscosity prevents singularity formation we also present for a certain class of initial conditions solutions of the equation with 0 1 3 and 1 that become infinite for 0 in finite time

the fractional laplacian in power weighted l p spaces integration by parts formulas and self adjointness abstract we consider the fractional laplacian operator s let s 0 1 on euclidean space and investigate the validity of the classical integration by parts formula that connects the l 2 r d scalar product between a function and its fractional laplacian to the nonlocal norm of the fractional sobolev space h s r d more precisely we focus on functions belonging to some weighted l 2 space whose fractional laplacian belongs to another weighted l 2 space we prove and disprove the validity of the integration by parts formula depending on the behaviour of the weight x at infinity the latter is assumed to be like a power both near the origin and at infinity the two powers being possibly different our results have direct consequences for the self adjointness of the linear operator formally given by 1 s the generality of the techniques developed allows us to deal with weighted l p spaces as well

simulated annealing via sobolev inequalities we use sobolev inequalities to study the simulated annealing algorithm this approach takes advantage of the local time reversibility of the process and yields the optimal freezing schedule as well as quantitative information about the rate at which the process is tending to its ground state

on numerical methods and error estimates for degenerate fractional convection diffusion equations first we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations secondly we develop a new kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods even when the principal derivatives have any fractional order between 1 and 2 the class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general levy diffusion special cases are conservation laws fractional conservation laws certain fractional porous medium equations and new strongly degenerate equations

formation of singularities for a transport equation with nonlocal velocity we study a 1d transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data by adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time

a general fractional porous medium equation we develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion u t 2 u m 1 u 0 x r n t 0 u x 0 f x x r n we consider data f l 1 r n and all exponents 0 0 existence and uniqueness of a weak solution is established for m m n n giving rise to an l 1 contraction semigroup in addition we obtain the main qualitative properties of these solutions in the lower range 0 m m existence and uniqueness of solutions with good properties happen under some restrictions and the properties are different from the case above m we also study the dependence of solutions on f m and moreover we consider the above questions for the problem posed in a bounded domain

a monotonicity formula and a liouville type theorem for a fourth order supercritical problem we consider liouville type and partial regularity results for the nonlinear fourth order problem

diffusion for global optimization in r n we seek a global minimum of u mathbb r n to mathbb r the solution to d dt x t nabla u x t will find local minima using the idea of simulated annealing we consider th

pattern formation in a flux limited reaction diffusion equation of porous media type a non linear pde featuring flux limitation effects together with those of the porous media equation non linear fokker planck is presented in this paper we analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns namely traveling wave solutions to this singular reaction diffusion equation we show the existence and qualitative behavior of different types of traveling waves classical profiles for wave speeds high enough and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold some of these solutions are of particular relevance as they provide models by which the whole solution and not just the bulk of it as it is the case with classical traveling waves spreads through the medium with finite speed

transformations of self similar solutions for porous medium equations of fractional type abstract we consider four different models of nonlinear diffusion equations involving fractional laplacians and study the existence and properties of classes of self similar solutions such solutions are an important tool in developing the general theory we introduce a number of transformations that allow us to map complete classes of solutions of one equation into those of another one thus providing us with a number of new solutions as well as interesting connections special attention is paid to the property of finite propagation

uniqueness and properties of distributional solutions of nonlocal equations of porous medium type abstract we study the uniqueness existence and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation t u l u 0 here l can be any nonlocal symmetric degenerate elliptic operator including the fractional laplacian and numerical discretizations of this operator the function r r is only assumed to be continuous and nondecreasing the class of equations include nonlocal generalized porous medium equations fast diffusion equations and stefan problems in addition to very general uniqueness and existence results we obtain stability l 1 contraction and a priori estimates we also study local limits continuous dependence and properties and convergence of a numerical approximation of our equations

the nonlocal porous medium equation barenblatt profiles and other weak solutions a degenerate nonlinear nonlocal evolution equation is considered it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal we show the existence of sign changing weak solutions to the corresponding cauchy problem moreover we construct explicit compactly supported self similar solutions which generalize barenblatt profiles the well known solutions of the classical porous medium equation

on a class of elliptic problems in r symmetry and uniqueness results in the plane r 2 we classify all solutions for an elliptic problem of liouville type involving a radial weight function as a consequence we clarify the origin of the non radially symmetric solutions for the given problem as established by chanillo and kiessling for a more general class of liouville type problems we show that rather than radial symmetry the solutions always inherit the invariance of the problem under inversion with respect to suitable circles this symmetry result is derived with the help of a shrinking sphere method

initial trace of positive solutions to fractional diffusion equation with absorption in this paper we prove the existence of an initial trace t u of any positive solution u of the semilinear fractional diffusion equation h t u u f t x u 0 in r times r n where n 1 where the operator with 0 1 is the fractional laplacian and f r times r n times r r is a caratheodory function satisfying f t x u u 0 for all t x u r times r n times r we define the regular set of the trace t u as an open subset of r u r n carrying a nonnegative radon measive u such that lim t 0 ru u t x x dx ru d c 2 0 r u and the singular set s u r n r u as the set points a such that lim sup t 0 b a u t x dx 0 we study the reverse problem of constructing a positive solution to h with a given initial trace s where s r n is a closed set and is a positive radon measure on r r n s and develop the case f t x u t u p where 1 and p 1

local lipschitz continuity of the inverse of the fractional p laplacian holder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains let p 1 s 0 1 and rn an arbitrary bounded open set in the first part we consider the inverse s p p s 1 of the fractional p laplace operator p s with the dirichlet boundary condition we show that in the singular case p 1 2 the operator s p is locally lipschitz continuous on l and that global lipschitz continuity cannot be achieved we use this result to show that in the case n sp if 2nn 2s p 2 0 q npn sp 2 and are small constants then the nonlinear problem psu u qu u npn sp 2u hin u 0 on rn has at least one weak solution in the second part of the paper we prove that the operator p s generates a nonlinear submarkovian semigroup ss p t t 0 on l2 if p 2 and sp 0 such that for every t 0 and u v lq q 2 we have ss p t u ss p t v l c s t s u v lq s where s s and s are explicit constants depending only on n s p q

classification and nondegeneracy of su n 1 toda system with singular sources we consider the following toda system open image in new window where i 1 0 is dirac measure at 0 and the coefficients a ij form the standard tri diagonal cartan matrix in this paper i we completely classify the solutions and obtain the quantization result sum j 1 n a ij int mathbb r 2 e u j dx 4 pi 2 gamma i gamma n 1 i quad forall 1 leq i leq n this generalizes the classification result by jost and wang for i 0 forall 1 leq i leq n ii we prove that if i i 1 j for all 1 i j n then any solution u i is radially symmetric w r t 0 iii we prove that the linearized equation at any solution is non degenerate these are fundamental results in order to understand the bubbling behavior of the toda system

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

smoothing effects for the filtration equation with different powers abstract we study the nonlinear diffusion equation u t u on general euclidean domains with homogeneous neumann boundary conditions we assume that u is bounded from below by u m 1 1 for small u and by u m 2 1 for large u the two exponents m 1 m 2 being possibly different and larger than one the equality case corresponds to the well known porous medium equation we establish sharp short and long time l q 0 l smoothing estimates similar issues have widely been investigated in the literature in the last few years but the neumann problem with different powers had not been addressed yet this work extends some previous results in many directions

the fisher kpp problem with doubly nonlinear diffusion abstract the famous fisher kpp reaction diffusion model combines linear diffusion with the typical kpp reaction term and appears in a number of relevant applications in biology and chemistry it is remarkable as a mathematical model since it possesses a family of travelling waves that describe the asymptotic behaviour of a large class solutions 0 u x t 1 of the problem posed in the real line the existence of propagation waves with finite speed has been confirmed in some related models and disproved in others we investigate here the corresponding theory when the linear diffusion is replaced by the slow doubly nonlinear diffusion and we find travelling waves that represent the wave propagation of more general solutions even when we extend the study to several space dimensions a similar study is performed in the critical case that we call pseudo linear i e when the operator is still nonlinear but has homogeneity one with respect to the classical model and the pseudo linear case the slow travelling waves exhibit free boundaries

the fractional schrodinger equation with general nonnegative potentials the weighted space approach abstract we study the dirichlet problem for the stationary schrodinger fractional laplacian equation s u v u f posed in bounded domain r n with zero outside conditions we consider general nonnegative potentials v l l o c 1 and prove well posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions f important properties of the solutions such as its boundary behaviour are derived the case of super singular potentials that blow up near the boundary is given special consideration since it leads to so called flat solutions we comment on related literature

duality on gradient estimates and wasserstein controls abstract we establish a duality between l p wasserstein control and l q gradient estimate in a general framework our result extends a known result for a heat flow on a riemannian manifold especially we can derive a wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound by applying our result to a subelliptic heat flow on a lie group we obtain a coupling of heat distributions which carries a good control of their relative distance

on nonlocal quasilinear equations and their local limits we introduce a new class of quasilinear nonlocal operators and study equations involving these operators the operators are degenerate elliptic and may have arbitrary growth in the gradient included are new nonlocal versions of p laplace laplace mean curvature of graph and even strongly degenerate operators in addition to some nonlocal quasilinear operators appearing in the existing literature our main results are comparison uniqueness and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators because of the structure of our operators especially the existence proof is highly non trivial and non standard we also identify the conditions under which the nonlocal operators converge to local quasilinear operators and show that the solutions of the corresponding nonlocal equations converge to the solutions of the local limit equations finally we give a formal stochastic representation formula for the solutions and provide many examples

fractional cahn hilliard allen cahn and porous medium equations abstract we introduce a fractional variant of the cahn hilliard equation settled in a bounded domain r n and complemented with homogeneous dirichlet boundary conditions of solid type i e imposed in the whole of r n after setting a proper functional framework we prove existence and uniqueness of weak solutions to the related initial boundary value problem then we investigate some significant singular limits obtained as the order of either of the fractional laplacians appearing in the equation is let tend to 0 in particular we can rigorously prove that the fractional allen cahn fractional porous medium and fractional fast diffusion equations can be obtained in the limit finally in the last part of the paper we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits

uniqueness of very weak solutions for a fractional filtration equation abstract we prove existence and uniqueness of distributional bounded solutions to a fractional filtration equation in r d with regards to uniqueness it was shown even for more general equations in 22 that if two bounded solutions u w of 1 1 satisfy u w l 1 r d 0 t then u w we obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions for nonnegative initial data we first show that a minimal solution exists and then that any other solution must coincide with it a similar procedure is carried out for sign changing solutions as a consequence distributional solutions have locally finite energy

weak solution of the equation for a fractional porous medium with a forcing term we consider the cauchy problem with a forcing term for a fractional porous medium which arises in statistical mechanics and heat control the existence and uniqueness of a weak energy solution are established using implicit time discretization and l 1 contraction semigroup arguments

asymptotics of the spectral gap with applications to the theory of simulated annealing soit m une variete de riemann compacte connexe a n dimensions de produit interieur et soit u m r 1 une fonction lisse donnee pour 0 on definit 7b l c m c m par 7b l o exp u exp u o ou et sont le gradient et la divergence sur m soit la taille de l ecart entre 0 et le reste du spectre de 7b l on etudie le comportement asymptotique de

a fractional porous medium equation abstract we develop a theory of existence uniqueness and regularity for the following porous medium equation with fractional diffusion u t 1 2 u m 1 u 0 x r n t 0 u x 0 f x x r n with m m n 1 n n 1 and f l 1 r n an l 1 contraction semigroup is constructed and the continuous dependence on data and exponent is established nonnegative solutions are proved to be continuous and strictly positive for all x r n t 0

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

continuity of the temperature in boundary heat control problems abstract motivated by the boundary heat control problems formulated in the book of duvaut and lions we study a boundary stefan problem and a boundary porous media problem we prove continuity of the solution with the appropriate modulus we also extend the results to the fractional order case and to the anomalous diffusion problems

the gel fand problem for the biharmonic operator we study stable and finite morse index solutions of the equation delta 2 u e u if the equation is posed in mathbb r n we classify radial stable solutions we then construct nonradial stable solutions and we prove that unlike the corresponding second order problem no liouville type theorem holds unless additional information is available on the asymptotics of solutions at infinity thanks to this analysis we prove that stable solutions of the equation on a smoothly bounded domain supplemented with navier boundary conditions are smooth if and only if n leqq 12 we find an upper bound for the hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded morse index provided they are controlled in a suitable morrey norm

on the well posedness of a spectral fractional forward backward pseudo parabolic equation abstract we prove existence uniqueness and stability of solution of a forward backward pseudo parabolic equation with spectral fractional laplacian then we define and prove existence of young measure valued solution for fractional forward backward problem by the vanishing viscosity method

poincare and logarithmic sobolev inequalities by decomposition of the energy landscape we consider a diffusion on a potential landscape which is given by a smooth hamiltonian h rn r in the regime of low temperature e we proof the eyring kramers formula for the optimal constant in the poincare pi and logarithmic sobolev inequality lsi for the associated generator l e h of the diffusion the proof is based on a refinement of the two scale approach introduced by grunewald otto westdickenberg and villani govw09 and of the mean difference estimate introduced by chafai and malrieu cm10 the eyring kramers formula follows as a simple corollary from two main ingredients the first one shows that the pi and lsi constant of the diffusion restricted to a basin of attraction of a local minimum scales well in e this mimics the fast convergence of the diffusion to metastable states the second ingredient is the estimation of a mean difference by a weighted transport distance it contains the main contribution to the pi and lsi constant resulting from exponentially long waiting times of jumps between metastable states of the diffusion

infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains we study the positivity and regularity of solutions to the fractional porous me dium equations ut sum 0 in 0 for m 1 and s 0 1 with dirichlet boundary data u 0 in 0 n and nonnegative initial condition u 0 u0 0 our first result is a quantitative lower bound for solutions that holds for all positive times t 0 as a consequence we find a global harnack principle stating that for any t 0 solutions are comparable to ds m where d is the distance to this is in sharp contrast with the local case s 1 where the equation has finite speed of propagation after this we study the regularity of solutions we prove that solutions are classical in the interior c in x and c1 in t and establish a sharp cxs m regularity estimate up to the boundary our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form ut f u 0 in both in bounded and unbounded domains 2016 wiley periodicals inc

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

the dirichlet problem for the fractional p laplacian evolution equation abstract we consider a model of fractional diffusion involving a natural nonlocal version of the p laplacian operator we study the dirichlet problem posed in a bounded domain of r n with zero data outside of for which the existence and uniqueness of strong nonnegative solutions is proved and a number of quantitative properties are established a main objective is proving the existence of a special separate variable solution u x t t 1 p 2 f x called the friendly giant which produces a universal upper bound and explains the large time behaviour of all nontrivial nonnegative solutions in a sharp way moreover the spatial profile f of this solution solves an interesting nonlocal elliptic problem we also prove everywhere positivity of nonnegative solutions with any nontrivial data a property that separates this equation from the standard p laplacian equation

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

partial regularity of finite morse index solutions to the lane emden equation we prove regularity and partial regularity results for finite morse index solutions u h1 lp to the lane emden equation u u p 1u in

analysis on nilpotent groups soit g un groupe de lie nilpotent reel simplement connexe et soit l son algebre de lie soient l 1 l et l k 1 l k l k 1 2 soit d d g d l k 1 kdim r l k l k 1 d est la dimension de dirichlet de g soit h un groupe nilpotent discret finiment genere on demontre des theoremes d analyse pour g

on the cauchy problem for a general fractional porous medium equation with variable density abstract we study the well posedness of the cauchy problem for a fractional porous medium equation with a varying density 0 we establish existence of weak energy solutions uniqueness and nonuniqueness is studied as well according to the behavior of at infinity

finite time extinction for stochastic sign fast diffusion and self organized criticality we prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space time noise corresponding to the bak tang wiesenfeld model for self organized criticality this solves a problem posed and left open in several works barbu methods appl sci 36 1726 1733 2013 rockner and wang j lond math soc 2 87 545 560 2013 barbu et al j math anal appl 389 147 164 2012 barbu and rockner comm math phys 311 539 555 2012 barbu et al comm math phys 285 901 923 2009 c r math acad sci paris 347 1 2 81 84 2009 the highly singular degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the analysis of mass diffusion and several new estimates and techniques are introduced

holder estimates for singular non local parabolic equations abstract in this paper we establish local holder estimate for non negative solutions of the singular equation m p below for m in the range of exponents n 2 n 2 1 since we have trouble in finding the local energy inequality of v directly we use the fact that the operator can be thought as the normal derivative of some extension v of v to the upper half space caffarelli and silvestre 2007 5 i e v is regarded as boundary value of v the solution of some local extension problem therefore the local holder estimate of v can be obtained by the same regularity of v in addition it enables us to describe the behavior of solution of non local fast diffusion equation near their extinction time

symmetrization for fractional neumann problems abstract in this paper we complement the program concerning the application of symmetrization methods to nonlocal pdes by providing new estimates in the sense of mass concentration comparison for solutions to linear fractional elliptic and parabolic pdes with neumann boundary conditions these results are achieved by employing suitable symmetrization arguments to the stinga torrea local extension problems corresponding to the fractional boundary value problems considered sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates is then established finally a parabolic symmetrization result is covered as an application of the elliptic concentration estimates in the implicit time discretization scheme

quantitative local and global a priori estimates for fractional nonlinear diffusion equations abstract we establish quantitative estimates for solutions u t x to the fractional nonlinear diffusion equation t u s u m 0 in the whole range of exponents m 0 0 s 1 the equation is posed in the whole space x r d we first obtain weighted global integral estimates that allow to establish existence of solutions for classes of large data in the core of the paper we obtain quantitative pointwise lower estimates of the positivity of the solutions depending only on the norm of the initial data in a certain ball the estimates take a different form in three exponent ranges slow diffusion good range of fast diffusion and very fast diffusion finally we show existence and uniqueness of initial traces

regularizing effects for ut aϑ u 0 in l1 abstract various initial boundary value problems and cauchy problems can be written in the form du dt a u 0 where r r is nondecreasing and a is the linear generator of strongly continuous nonexpansive semigroup e ta in an l1 space for example if a subject perhaps to suitable boundary conditions we obtain equations arising in flow in a porous medium or plasma physics depending on the choice of while if a x acting in l1 r we have a scalar conservation law in this paper we show that if m m 0 and m 2 m 2 where 1 1 then roughly speaking the norm of t du dt may be estimated in terms of the initial data u0 in l1 such estimates give information about the regularity of solutions asymptotic behaviour etc in applications side issues such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator a are also dealt with these considerations are of independent interest

optimal existence and uniqueness theory for the fractional heat equation abstract we construct a theory of existence uniqueness and regularity of solutions for the fractional heat equation t u s u 0 0 s 1 posed in the whole space r n with data in a class of locally bounded radon measures that are allowed to grow at infinity with an optimal growth rate we consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions which is given in one direction by the representation formula in the other one by the initial trace we review many of the typical properties of the solutions in particular we prove optimal pointwise estimates and new harnack inequalities

multiple end solutions to the allen cahn equation in r2 we construct a new class of entire solutions for the allen cahn equation u 1 u2 u 0 in r2 c given k 1 we find a family of solutions whose zero level sets are away from a compact set asymptotic to 2k straight lines which we call the ends these solutions have the property that there exist 0 1 2k 0 2 such that limr u rei 1 j uniformly in on compact subsets of j j 1 for j 0 2k 1