nonlinear degenerate parabolic equations with measure data abstract in this paper we study the existence of solution to nonlinear degenerate parabolic equations with measure data

initial trace of solutions of some quasilinear parabolic equations with absorption we study the existence of an initial trace of nonnegative solutions of the problem tu u p 2 u uq 0in qt 0 t we prove that the initial trace is an outer regular borel measure which may not be locally bounded for some values of the parameters p and q we study also the corresponding cauchy problems with a given generalized borel measure as initial data

renormalised solutions of nonlinear parabolic problems with l1 data existence and uniqueness in this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u 0 belong to l 1 0 t and l 1 and where the function a 0 t n n is monotone but not necessarily strictly monotone and defines a bounded coercive continuous operator from the space into its dual space the renormalised solution is an element of c 0 0 t l 1 such that its truncates t k u belong to with this solution satisfies the equation formally obtained by using in the equation the test function s u where belongs to and where s belongs to c with

irregular time dependent obstacles abstract we study the obstacle problem for the evolutionary p laplace equation when the obstacle is discontinuous and does not have regularity in the time variable two quite different procedures yield the same solution

on the existence of weak solutions for quasilinear parabolic initial boundary value problems structure conditions on a strongly nonlinear operator a u are given under whichthe initial dirichlet boundary value problem for has weak solutions

on some anisotropic reaction diffusion systems with l1 data modeling the propagation of an epidemic disease in this paper we prove the existence of weak solutions for a reaction di usion system with general anisotropic di usivities and transport e ects supplemented with either mixed boundary conditions or no 5ux boundary conditions initial conditions and external forcing terms are in l 1 this does not allow us to use classical variational formulations our motivation is a mathematical model describing the spatial propagation in heterogeneous environments of feline immunode7ciency virus a feline retrovirus 2003 elsevier science ltd all rights reserved msc 35k57 35k55 92d30

global aspects of 0 2 moduli space toric varieties and tangent bundles we study the moduli space of a 2 half twisted gauged linear sigma models for nef fano toric varieties focusing on toric deformations of the tangent bundle we describe the vacuum structure of many 0 2 theories in particular identifying loci in parameter space with spontaneous supersymmetry breaking or divergent ground ring correlators we find that the parameter space of such an a 2 theory and its ground ring is in general a moduli stack and we show in examples that with suitable stability conditions it is possible to obtain a simple compactification of the moduli space of smooth a 2 theories

existence uniqueness and behavior of solutions for a class of nonlinear parabolic problems abstract we prove existence uniqueness regularity results and estimates describing the behavior both for large and small times of a solution u of some nonlinear parabolic equations of leray lions type including the p laplacian in particular we show how the summability of the initial datum u 0 and the value of p influence the behavior of the solution u producing ultracontractive or supercontractive estimates or extinction in finite time or different kinds of decay estimates

parabolic equations with nonlinear singularities abstract we show the existence of positive solutions u l 2 0 t h 0 1 for nonlinear parabolic problems with singular lower order terms of the asymptote type more precisely we shall consider both semilinear problems whose model is u t u u 1 u f x t in 0 t u x 0 u 0 x in u x t 0 on 0 t and quasilinear problems having natural growth with respect to the gradient whose model is u t u u 2 u f x t in 0 t u x 0 u 0 x in u x t 0 on 0 t with 0 moreover we prove a comparison principle and as an application we study the asymptotic behavior of the solution as t goes to infinity

nonlinear parabolic equations with measure data in this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model isu div u p 2 u on 0 t p with homogeneous cauchy dirichlet boundary conditions wherep 1 and is a bounded measure on 0 t we also study how the summability of the gradient improves if the measure is a function inlm 0 t withm small moreover we give a new proof of the existence of a solution for problem p

pointwise gradient estimates abstract we survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear possibly degenerate elliptic and parabolic equations

well posedness of parabolic equations in the non reflexive and anisotropic musielak orlicz spaces in the class of renormalized solutions abstract we prove existence and uniqueness of renormalized solutions to general nonlinear parabolic equation in musielak orlicz space avoiding growth restrictions namely we consider t u div a x u f l 1 t on a lipschitz bounded domain in r n the growth of the weakly monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose on m is continuity of log holder type which results in good approximation properties of the space however the requirement of regularity can be skipped in the case of reflexive spaces the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments uniqueness results from the comparison principle

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

borderline gradient continuity for nonlinear parabolic systems we consider the evolutionary p laplacean system begin aligned partial t u triangle p u f qquad p frac 2n n 2 end aligned in cylindrical domains of mathbb r n times mathbb r and prove the continuity of the spatial gradient du under the lorentz space assumption f in l n 2 1 when f is time independent the condition improves in f in l n 1 this is the limiting case of a result of dibenedetto claiming that du is holder continuous when f in l q for q n 2 at the same time this is the natural nonlinear parabolic analog of a linear result of stein claiming the gradient continuity of solutions to the linear elliptic system triangle u in l n 1 is continuous new potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions

the low mach number limit for the full navier stokes fourier system we study the low mach number asymptotic limit for solutions to the full navier stokes fourier system supplemented with ill prepared data and considered on an arbitrary time interval convergencetowards the incompressible navier stokes equations is shown

p laplacian with a concentrated nonlinear source abstract in this paper we consider the cauchy problem for the p laplacian with a concentrated nonlinear source we obtain the existence of the generalized solutions for the problem based on some a priori estimates on solutions

nonlinear parabolic capacity and polar sets of superparabolic functions we extend the theory of the thermal capacity for the heat equation to nonlinear parabolic equations of the p laplacian type we study definitions and properties of the nonlinear parabolic capacity and show that the capacity of a compact set can be represented via a capacitary potential as an application we show that polar sets of superparabolic functions are of zero capacity the main technical tools used include estimates for equations with measure data and obstacle problems

insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures this is the peer reviewed version of the following article ruiz girones e oliver a socorro g cascon j escobar j m montenegro r sarrate j insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures international journal for numerical methods in engineering 2018 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 5706 pdf this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

singular parabolic equations measures satisfying density conditions and gradient integrability abstract we consider solutions to singular parabolic equations with measurable dependence on the x t variables and having on the right hand side a measure satisfying a density condition we prove that the less the measure is concentrated the more the gradient is regular in the marcinkiewicz scale we provide local estimates and recover some classic results

strong regularizing effect of a gradient term in the heat equation with the hardy potential abstract we deal with the following parabolic problem u t u u p u x 2 f u 0 in 0 t u x t 0 on 0 t u x 0 u 0 x x where r n n 3 is a bounded regular domain such that 0 or r n 1 p 2 0 and f 0 u 0 0 are in a suitable class of functions for p p n n 1 we will show that the above problem has a solution for all 0 f l 1 t and u 0 l 1 we prove also that p is optimal for the existence result these results prove the strong regularizing effect of a gradient term in the problem studied in baras and goldstein 1984 3 the cauchy problem is also studied

adaptive finite element method for parabolic equations with dirac measure abstract in this paper we study the adaptive finite element method for parabolic equations with dirac measure two kinds of problems with separate measure data in time and measure data in space are considered it is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the dirac measure and thus fit to adaptive fem for space discretization and variable time steps for time discretization for both cases we use piecewise linear and continuous finite elements for the space discretization and backward euler scheme or equivalently piecewise constant discontinuous galerkin method for the time discretization the a posteriori error estimates based on energy and l 2 norms for the fully discrete problems are then derived to guide the adaptive procedure numerical results are provided at the end of the paper to support our theoretical findings

riesz potentials and nonlinear parabolic equations the spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric riesz potential of the right hand side datum exactly as in the case of the heat equation heat kernels type estimates persist in the nonlinear case

convergence of a finite volume scheme for the convection diffusion equation with l 1 data in this paper we prove the convergence of a finite volume scheme for the time dependent convection diffusion equation with an l 1 right hand side to this purpose we first prove estimates for the discrete solution and for its discrete time and space derivatives then we show the convergence of a sequence of discrete solutions obtained with more and more refined discretiza tions possibly up to the extraction of a subsequence to a function which mets the regularity requirements of the weak formulation of the problem to this purpose we prove a compactness result which may be seen as a discrete analogue to aubin simon s lemma finally such a limit is shown to be indeed a weak solution

some elliptic equations with w01 1 solutions abstract we consider some nonlinear dirichlet problems and we study how lower order terms can give a regularizing effect on the solutions the existence of distributional solutions with minimal properties solutions in w 0 1 1 functional space not so usual for finding solutions of elliptic problems or finite energy solutions even with nonregular data

moser nash kernel estimates for degenerate parabolic equations abstract in this paper we deal with the cauchy problem associated to a class of nonlinear degenerate parabolic equations whose prototype is the parabolic p laplacian 2 p in his seminal paper after stating the harnack estimates moser proved almost optimal estimates for the parabolic kernel by using the so called harnack chain method in the linear case sharp estimates come by using nash s approach fabes and stroock proved that gaussian estimates are equivalent to a parabolic harnack inequality in this paper by using the dibenedetto degiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations to obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions

nonlinear parabolic problems with lower order terms and related integral estimates abstract we deal with the solutions to nonlinear parabolic equations of the form u t div a x t d u g x t u f x t on t t 0 under standard growth conditions on g and a with f only assumed to be integrable to the power 1 we prove general local decay estimates for level sets of the solutions u and the gradient d u which imply very general estimates in rearrangement function spaces lebesgue orlicz lorentz and non rearrangement ones up to lorentz morrey spaces

w01 1 solutions in some borderline cases of calderon zygmund theory abstract in this paper we study the existence of w 0 1 1 distributional solutions of dirichlet problems whose simplest example is div u p 2 u f x in u 0 on

on the coupling between a navier stokes equation and a total energy equation in spatial dimension n 3 abstract we consider a class of nonlinear evolution systems namely the rayleigh benard equations this system arises from the coupling between a navier stokes equation for the velocity and the pressure and a total energy equation in spatial dimension n 3 we give a few existence results of solutions under suitable conditions in the right hand side of the momentum equation the forcing term depending on the temperature to this end we begin to solve an approximated problem namely the boussinesq system resulting from the rayleigh benard equations through a fixed point argument next by a linear combination we construct a new equivalent system finally we give a priori estimates and compactness results before passing to the limit in the equivalent system

renormalized solutions to nonlinear parabolic problems in generalized musielak orlicz spaces abstract we will present the proof of existence of renormalized solutions to a nonlinear parabolic problem t u div a d u f with right hand side f and initial data u 0 in l 1 the growth and coercivity conditions on the monotone vector field a are prescribed by a generalized n function m which is anisotropic and inhomogeneous with respect to the space variable in particular m does not have to satisfy an upper growth bound described by a 2 condition therefore we work with generalized musielak orlicz spaces which are not necessarily reflexive moreover we provide a weak sequential stability result for a more general problem t u div a d u f u f where is a monotone function with respect to the second variable and f is locally lipschitz continuous within the proof we use truncation methods young measure techniques the integration by parts formula and monotonicity arguments which have been adapted to nonreflexive musielak orlicz spaces

existence results for a cauchy dirichlet parabolic problem with a repulsive gradient term abstract we study the existence of solutions of a nonlinear parabolic problem of cauchy dirichlettype having a lower order term which depends on the gradient the model we have in mind is the following u t div a t x u u p 2 u q f t x in q t u 0 on 0 t u 0 x u 0 x in where q t 0 t is a bounded domain of r n n 2 1 p n the matrix a t x is coercive and with measurable bounded coefficients the r h s growth rate satisfies the superlinearity condition max p 2 p n 1 n n 2 q p and the initial datum u 0 is an unbounded function belonging to a suitable lebesgue space l we point out that once we have fixed q there exists a link between this growth rate and exponent q n p which allows one to have or not an existence result moreover the value of q deeply influences the notion of solution we can ask for the sublinear growth case with 0 q p 2 is dealt at the end of the paper for what concerns small value of p namely 1 p 2

weak solutions to fokker planck equations and mean field games we deal with systems of pdes arising in mean field games theory where viscous hamilton jacobi and fokker planck equations are coupled in a forward backward structure we consider the case of local coupling when the running cost depends on the pointwise value of the distribution density of the agents in which case the smoothness of solutions is mostly unknown we develop a complete weak theory proving that those systems are well posed in the class of weak solutions for monotone couplings under general growth conditions and for superlinear convex hamiltonians as a key tool we prove new results for fokker planck equations under minimal assumptions on the drift through a characterization of weak and renormalized solutions the results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

global attractors for the p laplacian equations with nonregular data abstract let be a smooth bounded domain in r n n 3 we consider the long time behavior of solutions to the p laplacian equation u t p u f u g where 2 p n p div u p 2 u is the p laplace operator assume that g and the initial condition u 0 lie in l 1 f r r is of class c 1 and satisfies proper growth conditions firstly we prove the uniqueness of the entropy solution and establish some regularity results then we show the existence of a global attractor a in l r 1 w 0 1 s with s max n p 1 n 1 p r 1 r to obtain the results a decomposition method and a bootstrap technique are used

dgm a deep learning algorithm for solving partial differential equations abstract high dimensional pdes have been a longstanding computational challenge we propose to solve high dimensional pdes by approximating the solution with a deep neural network which is trained to satisfy the differential operator initial condition and boundary conditions our algorithm is meshfree which is key since meshes become infeasible in higher dimensions instead of forming a mesh the neural network is trained on batches of randomly sampled time and space points the algorithm is tested on a class of high dimensional free boundary pdes which we are able to accurately solve in up to 200 dimensions the algorithm is also tested on a high dimensional hamilton jacobi bellman pde and burgers equation the deep learning algorithm approximates the general solution to the burgers equation for a continuum of different boundary conditions and physical conditions which can be viewed as a high dimensional space we call the algorithm a deep galerkin method dgm since it is similar in spirit to galerkin methods with the solution approximated by a neural network instead of a linear combination of basis functions in addition we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic pdes

nonlinear potential theory of elliptic systems abstract we give a summary of recent results from nonlinear potential theory focusing on linear and nonlinear potential estimates of solutions to non homogeneous equations and systems we start with the cases of quasilinear possibly degenerate equations of p laplacian type and passing through fully nonlinear elliptic equations finally move to systems in this last case we describe recent results implying potential estimates for the p laplacian system finally we describe results bridging nonlinear potential theory and classical partial regularity theory for general elliptic systems the main outcomes are new e regularity criteria involving both excess functionals and nonlinear potentials

existence of solutions for degenerate parabolic equations with singular terms abstract in this paper we deal with parabolic problems whose simplest model is u p u b u p u 0 in 0 t u 0 x u 0 x in u t x 0 on 0 t where t 0 n 2 p 1 b 0 and u 0 is a positive function in l bounded away from zero

optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a hardy potential abstract we deal with the following parabolic problem u t u u p u x 2 f u 0 in 0 t u x t 0 on 0 t u x 0 u 0 x x where r n n 3 is a bounded regular domain such that 0 or r n p 1 0 and f 0 u 0 0 are in a suitable class of functions there are deep differences with respect to the heat equation 0 the main features in the paper are the following if 0 there exists a critical exponent p such that for p p there is no nontrivial local solution p is optimal in the sense that if p p there exists solution for suitable data if we consider the cauchy problem i e r n we find the same phenomenon about the critical power p as above moreover there exists a fujita type exponent f p in the sense that independently of the initial datum for 1 p f any solution blows up in a finite time respect to an integral norm this is a major difference with respect to the heat equation 0

renormalized solutions to parabolic equations in time and space dependent anisotropic musielak orlicz spaces in absence of lavrentiev s phenomenon abstract the paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a lipschitz bounded domain in r n our focal point is to involve the leading nonlinear part of the operator whose growth is described by anisotropic n function m inhomogeneous in the space and the time variables the main goals are proven in absence of lavrentiev s phenomenon to ensure which we impose a certain type of balance of interplay between the behavior of m for large and small changes of time and space variables its instances are log holder continuity of variable exponent inhomogeneous in time and space or optimal closeness condition for powers in double phase spaces possibly changing in time new delicate approximation in time result is proven and applied in the construction of renormalized solutions

a nonlinear parabolic problem with singular terms and nonregular data abstract we study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum of the form u t p u h u f in 0 t u 0 on 0 t u u 0 in 0 where is an open bounded subset of r n n 2 u 0 is a nonnegative integrable function p is the p laplace operator is a nonnegative bounded radon measure on 0 t and f is a nonnegative function of l 1 0 t the term h is a positive continuous function possibly blowing up at the origin furthermore we show uniqueness of finite energy solutions in presence of a nonincreasing h

a non existence result for nonlinear parabolic equations with singular measures as data in this paper we prove a non existence result for nonlinear parabolic problems with zero lower order terms whose model is where p div u p 2 u is the usual p laplace operator is measure concentrated on a set of zero parabolic r capacity 1 p r and q is large enough

fractional differentiability for solutions of a class of parabolic systems with l1 θ data abstract we study the higher differentiability of the very weak solution of the cauchy dirichlet problem associated to the linear parabolic system of the form u t div a x t d u in 0 t u 0 on 0 t u 0 on 0 under a weak holder continuity condition on the coefficients and the right hand side belonging to the morrey space l 1 in such a way we extend to the vectorial parabolic setting the result by di castro and palatucci 2013

a predator prey system with l1 data in this paper we are concerned with a system of nonlinear partial differential equations modeling a predator prey system in heterogeneous habitats preys are assumed to follow a logistic growth in the absence of predation and predators are assumed to feed on preys with a holling type ii functional response to prey density also interactions between predators are modelized by the statement of a food pyramid condition assuming no flux boundary conditions and l 1 data we prove the existence of at least one weak solution

renormalized solutions to a system of type navier stokes abstract the aim of this paper is to show the existence of weak renormalized solutions to a system of type navier stokes in the two dimensional case this system may be regarded as a modified version of the non isothermal solidification problem with melt convection

global weighted orlicz estimates for parabolic measure data problems application to estimates in variable exponent spaces abstract we consider a nonlinear parabolic equation with measurable nonlinearity in a nonsmooth bounded domain when the right hand side is a finite signed radon measure under optimal regularity assumptions on the nonlinearity and the boundary of the domain we prove a global calderon zygmund type estimate in weighted orlicz spaces as an application we obtain such an estimate in variable exponent spaces which gives an alternative proof for this new result in the literature

unbounded supersolutions of some quasilinear parabolic equations a dichotomy abstract we study unbounded supersolutions of the evolutionary p laplace equation with slow diffusion they are the same functions as the viscosity supersolutions a fascinating dichotomy prevails either they are locally summable to the power p 1 n p 0 or not summable to the power p 2 there is a void gap between these exponents those summable to the power p 2 induce a radon measure while those of the other kind do not we also sketch similar results for the porous medium equation

calderon zygmund estimates for parabolic measure data equations abstract we consider parabolic equations of the type u t div a x t d u having a radon measure on the right hand side and prove fractional integrability and differentiability results of calderon zygmund type for weak solutions we extend some of the integrability results for elliptic equations achieved by g mingione 2007 24 to the parabolic setting and locally recover the integrability results of l boccardo a dall aglio t gallouet and l orsina 1997 in 5

obstacle problem for a class of parabolic equations of generalized p laplacian type abstract we study nonlinear parabolic pdes with orlicz type growth conditions the main result gives the existence of a unique solution to the obstacle problem related to these equations to achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution moreover we prove that if the obstacle is continuous so is the solution