regularity in parabolic systems with continuous coefficients under the only assumption of continuous coefficients we prove a partial holder continuity result for solutions to parabolic systems with polynomial growth a key component throughout the argument is the use of dibenedetto s intrinsic geometry degenerate parabolic equations universitext springer new york 1993 to accommodate the inhomogeneity in the system a main technical point is that although we are proving the holder continuity of solutions we employ the intrinsic geometry using cylinder stretched according to the size of the spatial gradient in an iteration scheme that does not necessarily imply the boundedness of the gradient itself

existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media we prove the existence of multidimensional traveling wave solutions to the scalar equation for the transport of solutes contaminants with nonlinear adsorption and spatially periodic convection diffusion adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the lax and oleinik entropy conditions in the nondegenerate case we also prove the uniqueness of the traveling waves these traveling waves are analogues of viscous shock profiles they propagate with effective speeds that depend on the periodic porous media only up to their mean states and are given by an averaged rankine hugoniot relation this is a direct consequence of the fact that the transport equation is in conservation form we use the sliding domain method the continuation method spectral theory maximum principles and a priori estimates in the degenerate case the traveling waves are weak solutions of a degenerate parabolic equation and are only holder continuous we obtain them by taking suitable limits on the non degenerate traveling waves the uniqueness of the degenerate traveling waves is open

on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

holder continuity of parabolic quasi minimizers abstract we are concerned with scalar valued local parabolic quasi minimizers u associated to a caratheodory integrand f obeying p growth assumptions for p 1 for such parabolic quasi minimizers we prove local holder continuity provided they are already locally bounded the superquadratic case p 2 has already been considered in 17 22 whereas the subquadratic case could at least to our knowledge not be treated in the past in this paper we are able to handle both the sub and superquadratic case under more general growth conditions

non negative solutions of the evolution p laplacian equation initial traces and cauchy problem when 1 p 2 from the introduction we continue here the investigation initiated earlier by us trans amer math soc 314 1989 no 1 187 224 concerning the solvability of the cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation 1 ut div du p 2 du 0 in st rn 0 t 0 2 both in terms of results and techniques the main difference stems from the fact that unlike solutions in the case p 2 solutions of 1 are not in general locally bounded

very weak solutions of degenerate parabolic systems with non standard p x t growth abstract we study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p x t laplacian system t u div d u p x t 2 d u div f p x t 2 f under natural assumptions on the exponent function p 0 t 2 we prove that any very weak solution u 0 t r n with d u p 1 e l 1 belongs to the natural energy space i e d u p l loc 1 provided e 0 is small enough

remarks on holder continuity for solutions of the p laplace evolution equations we study the interior holder regularity problem for the gradient of solutions of the p laplace evolution equations with the external forces misawa gave some conditions for the holder continuity of the gradient of solutions we show holder estimates of the solutions with weaker condition as for misawa

local regularity for quasi linear parabolic equations in non divergence form abstract we consider viscosity solutions to non homogeneous degenerate and singular parabolic equations of the p laplacian type and in non divergence form we provide local holder and lipschitz estimates for the solutions in the degenerate case we prove the holder regularity of the gradient our study is based on a combination of the method of alternatives and the improvement of flatness estimates

regularity results for nonlinear parabolic obstacle problems with subquadratic growth abstract in 26 it was shown that the spatial gradient of the solution u to the parabolic obstacle problem with superquadratic growth is local holder continuous provided the obstacle is regular enough in this paper we extend this regularity result to the subquadratic case this means we establish the local holder continuity of the spatial gradient of the solution u to the parabolic obstacle problem with subquadratic growth more precisely we prove that d u c loc 0 2 for some 0 1 provided the coefficients and the obstacle are regular enough moreover we use the local holder continuity to prove the local lipschitz continuity of the solution u i e u c loc 0 1 1 2

an algebraic study of an extension of a class of quadrature formula abstract an extension of a class of quadrature formulae of the type first introduced by kronrod is considered in particular we examine properties of the nodes and the signs of the weights several conjectures are suggested by the numerical results

stopping functionals for gaussian quadrature formulas abstract gaussian formulas are among the most often used quadrature formulas in practice in this survey an overview is given on stopping functionals for gaussian formulas which are of the same type as quadrature formulas i e linear combinations of function evaluations in particular methods based on extended formulas like the important gauss kronrod and patterson schemes and methods which are based on gaussian nodes are presented and compared

well posedness and gradient blow up estimate near the boundary for a hamilton jacobi equation with degenerate diffusion abstract this paper is concerned with weak solutions of the degenerate diffusive hamilton jacobi equation t u p u u q with dirichlet boundary conditions in a bounded domain r n where p 2 and q p 1 with the goal of studying the gradient blow up phenomenon for this problem we first establish local well posedness with blow up alternative in w 1 norm we then obtain a precise gradient estimate involving the distance to the boundary it shows in particular that the gradient blow up can take place only on the boundary a regularizing effect for t u is also obtained

local continuity and asymptotic behaviour of degenerate parabolic systems abstract we study the local holder continuity and the asymptotic behaviour of solution u u 1 u k of the degenerate system u t i m u m 1 u i for m 1 and i 1 k which describes the population densities of k species whose diffusions are determined by their total population density u u 1 u k for the local holder continuity we adopt the intrinsic scaling and iteration arguments of degiorgi moser and dibenedetto under some regularity conditions we also prove that the population density of i th species with the population m i converges in c s to the function m i m b m x t as t where b m is the barenblatt profile of the standard porous medium equation with l 1 mass m m 1 m k as a consequence of asymptotic behaviour it is shown that each density becomes a concave function after a finite time

on the cauchy problem for the evolution p laplacian equations with gradient term and source abstract in this paper the authors study the equation u t div d u p 2 d u u q 1 u d u l in r n with p 2 we first prove that for 1 l p 1 the solution exists at least for a short time then for p 2 l p 1 the existence and nonexistence of global in time solutions are studied in various situations

parabolic equations with vmo coefficients in sobolev spaces with mixed norms an lq lp theory of divergence and non divergence form parabolic equations is presented the main coefficients are supposed to belong to the class vmox which in particular contains all measurable functions depending only on t the method of proving simplifies the methods previously used in the case p q

boundedness of the gradient for a doubly nonlinear parabolic equation we prove the local boundedness of the gradient for positive solutions to a doubly nonlinear parabolic equation in the case when the standard lebesgue measure has been replaced by a doubling measure which supports a weak poincare inequality

the number of peaks of nonnegative solutions to some nonlinear degenerate parabolic equations abstract we consider the cauchy problems for some nonlinear degenerate parabolic equations in r n for nonzero bounded nonnegative initial data having compact support and show that the set of peaks of the nonnegative solution consists of one point after a finite time

self improving property of the fast diffusion equation abstract we show that the gradient of the m power of a solution to a singular parabolic equation of porous medium type also known as fast diffusion equation satisfies a reverse holder inequality in suitable intrinsic cylinders relying on an intrinsic calderon zygmund covering argument we are able to prove the local higher integrability of such a gradient for m n 2 n 2 1 our estimates are satisfied for a general class of growth assumptions on the non linearity in this way we extend the theory for m 1 see 16 in the list of references to the singular case in particular an intrinsic metric that depends on the solution itself is introduced for the singular regime

partial regularity for subquadratic parabolic systems under controllable growth conditions in this paper we prove a regularity result for weak solutions away from singular set of parabolic systems with subquadratic growth condition of the type tu z diva z u du b z u du for z x t where the structure function a satisfies standard ellipticity and growth condition with polynomial growth rate 2nn 2 p 2 the inhomogeneity b satisfies the controllable growth condition b z u du 1 u p 1 du p 1 1p where p n 2 pn 2 p if n 2 p here p can be any exponent if n 2 p the proof is based on the a caloric approximation lemma to parabolic systems with subquadratic growth

existence of weak solutions of doubly nonlinear parabolic equations abstract we deal with a cauchy dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space time cylinder for doubly nonlinear parabolic equations whose prototype is t u div u m 1 d u p 2 d u f with a non negative lebesgue function f on the right hand side where p 2 n n 2 and m 0 the central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f the constructed solution is obtained by a limit of approximations i e we use solutions of regularized cauchy dirichlet problems and pass to the limit to receive a solution for the original cauchy dirichlet problem

regularity for parabolic systems of uhlenbeck type with orlicz growth abstract we study the local regularity of p caloric functions or more generally of caloric functions in particular we study local solutions of non linear parabolic systems with homogeneous right hand side where the leading terms have uhlenbeck structure of orlicz type this paper closes the gap of 23 where liebermann proved that if the gradient of a solution is bounded it is holder continuous the crucial step is a novel local estimates for the gradient of the solutions which generalize and improve the pioneering estimates of dibenedetto and friedman 12 10 for the p laplace heat equation

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

existence for minimization in banach space with some applications abstract existence of a minimum is shown for a coercive isc differentiable function restricted to a finite intersection of half spaces under a structural assumption on the g teaux derivative this abstract existence result is illustrated by two examples from optimal control theory for distributed systems

the obstacle problem for the porous medium equation we prove existence results for the obstacle problem related to the porous medium equation for sufficiently regular obstacles we find continuous solutions whose time derivative belongs to the dual of a parabolic sobolev space we also employ the notion of weak solutions and show that for more general obstacles such a weak solution exists the latter result is a consequence of a stability property of weak solutions with respect to the obstacle

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

geometric c1 α regularity estimates for nonlinear evolution models abstract in this survey we establish geometric c 1 regularity estimates for bounded solutions of a number of nonlinear evolution models in divergence and non divergence form the main insights to obtain such estimates are based on geometric tangential methods and make use of systematic oscillation mechanisms combined with intrinsic scaling techniques

bounded solutions to a quasilinear and singular parabolic equation with p laplacian abstract in this paper we study the existence of a positive local in time solution for the following singular nonlinear problem with homogeneous dirichlet boundary conditions t u p u u f x u u in 0 t q t u 0 on 0 t u 0 in q t u 0 x u 0 0 in where stands for a regular bounded domain of r n p u is the p laplacian defined by p u div u p 2 u 2 p 0 and t 0 the nonlinear term f r r n r is a caratheodory function satisfying the growth condition f x s a s q 1 b c p p q for a a x s r and m where a c m 0 and b 0 are some constants and q p p where p p n n p if p n and p if p n we prove for any initial nonnegative data u 0 l r with r 2 large enough the existence of at least one weak solution to p in the case 2 1 p 1 we prove the uniqueness of the solution and further regularity results for that we use some estimates based on logarithmic sobolev inequalities to get ultracontractivity of a closely related semi group of linear operators

life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values abstract in this paper we investigate the behavior of the positive solution of the following cauchy problem u t div u m p 2 u m u q with initial value decaying at infinity and give a new secondary critical exponent for the existence of global and nonglobal solutions furthermore the large time behavior and the life span of solutions are also studied

higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth abstract the aim of this paper is to derive the self improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p x t nonstandard growth more precisely we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator

an algebraic study of gauss kronrod quadrature formulae for jacobi weight functions we study gauss kronrod quadrature formulae for the jacobi weight function w d t 1 t a 1 t and its special case a f a of the gegenbauer weight function we are interested in delineating regions in the a f plane resp intervals in a for which the quadrature rule has a the interlacing property i e the gauss nodes and the kronrod nodes interlace b all nodes contained in 1 1 c all weights positive d only real nodes not necessarily satisfying a and or b we determine the respective regions numerically for n 1 1 20 4 40 in the gegenbauer case and for n 1 1 10 in the jacobi case where n is the number of gauss nodes algebraic criteria in particular the vanishing of appropriate resultants and discriminants are used to determine the boundaries of the regions identifying properties a and d the regions for properties b and c are found more directly a number of conjectures are suggested by the numerical results finally the gauss kronrod formula for the weight w al 2 is obtained from the one for the weight w ca and similarly the gauss kronrod formula with an odd number of gauss nodes for the weight function w t itlv 1 t2 c is derived from the gauss kronrod formula for the weight w a l y 2

cauchy problem for fast diffusion equations with a gradient term we study the cauchy problem for fast diffusion equation with a gradient term of the form ut um duq where 0 0 and 0 0 however as q m the optimality of the stipulated assumptions is left open

c1 α local regularity of weak solutions of degenerate elliptic equations abstract it is demonstrated the local c 1 alpha nature of weak solutions of elliptic equations of the type 1 1 in the introduction under the degeneracy or singularity assumptions a sub 1 a sub 3

lorentz estimates for obstacle parabolic problems we prove that the spatial gradient of variational solutions to parabolic obstacle problems of p laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of lorentz spaces

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

cauchy problem for nonlinear parabolic equations with a gradient term we consider the cauchy problem of the nonlinear diffusive hamilton jacobi equation ut um duq where m 0 the existence and nonexistence of local and global solutions are studied by a priori estimates and compactness methods we prove that the cauchy problem of this equation has no nontrivial nonnegative global solutions if 0 moreover we obtain the global existence of solutions in the limiting case and the exact l estimates for local and global solutions

degenerate parabolic equations with initial data measures we address the problem of existence of solutions to degenerate and nondegenerate parabolic equations under optimal assumptions on the initial data which are assumed to be measures the requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation and with the form of the nonlinear forcing term the latter depends on the space gradient of a power of the solution applications to related problems are also outlined

the heat flow in nonlinear hodge theory we study the nonlinear hodge systemd 0 and 2 0 for an exterior form on a compact oriented riemannian manifold m where q is a given positive function the solutions are called harmonic forms they are the stationary points on cohomology classes of the functional e me 2 dm with e q q 2 the codifferential of a form is defined as 1 with 2 we evolve a given closed form 0 by the nonlinear heat flow system d for a time dependent exterior form x t on m this system is the differential of the normalized gradient flow u for e with 0 du under a technical assumption on the function 2 q q q we show that the nonlinear heat flow system d with initial condition 0 0 has a unique solution for all times which converges to a harmonic form in the cohomology class of 0 this yields a nonlinear hodge theorem that every cohomology class of m has a unique harmonic representative

holder estimates for parabolic p x t laplacian systems in this paper we establish the local holder continuity of the spatial gradient of weak solutions to the parabolic p x t laplacian system begin array lll partial t u rm div left a x t du p x t 2 du right 0 end array more precisely we prove that begin array lll du in c rm loc 0 alpha alpha 2 quad rm for some alpha in 0 1 end array provided p and a are holder continuous

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

regularity of weak solutions and supersolutions to the porous medium equation abstract we investigate the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation in particular we establish the equivalence of the conditions u m l loc 2 0 t h loc 1 and u m 1 2 l loc 2 0 t h loc 1 in the definition of weak solutions our proof is based on approximation by solutions to obstacle problems

local calderon zygmund estimates for parabolic minimizers abstract we establish local calderon zygmund estimates for parabolic minimizers u to certain variational inequalities the formal corresponding parabolic partial differential equation would be t u div f z u d u u f z u d u div f p 2 f h with p 2 n n 2 however since we do not impose any condition on the existence of the derivative u f the pde above may not have any meaning at all for proving the calderon zygmund estimates we thus remain solely on the level of parabolic minimizers precisely for q p we prove that d u l loc q if u is locally holder continuous and f l loc q and h l loc q for a q depending on q

travelling waves in a convection diffusion equation abstract we study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1 d euclidean space the diffusion coefficient depends on the gradient in analogy with the p laplacian and may be degenerate unconditional stability is established with respect to initial data perturbations in l 1 r

existence and uniqueness of optimal controls for a quasilinear parabolic equation we consider the quasilinear parabolic equation dot y bf a y bf f y varphi on q 0 t times omega viewing varphi as a control we seek to minimize j varphi varphi hat varphi 2 lambda left y hat y 2 mu y t hat eta 2 under suitable hypotheses it is shown that one has existence of an optimal control varphi and that this satisfies the appropriate optimality system further for small data j min j is small enough one has global uniqueness and continuous dependence on the data

holder zygmund estimates for degenerate parabolic systems abstract we consider energy solutions of the inhomogeneous parabolic p laplacian system t u div d u p 2 d u div g we show in the case p 2 that if the right hand side g is locally in l bmo then u is locally in l c 1 where c 1 is the 1 holder zygmund space this is the borderline case of the calderon zygmund theory we provide local quantitative estimates we also show that finer properties of g are conserved by du e g holder continuity moreover we prove a new decay for gradients of p caloric solutions for all 2 n n 2 p

the asymptotic behaviour of solutions of a porous medium equation with bounded measurable coefficients abstract we prove the holder continuity of the interface of a solution of a porous medium equation with bounded measurable coefficients we also describe the asymptotic behaviour as time goes to infinity

weak solutions of the porous medium equation we show that if u 0 u l loc m r n 1 solves u t u m m 1 in the sense of distributions then u is locally holder continuous in

a transport model with micro and macro structure abstract in this paper we consider solute transport through porous media where the chemical species undergoes a chemical process through the surface of the porous skeleton the problem is modeled by a system of differential equations for the macro concentration u u x t and the micro concentration u u x x t we prove existence and uniqueness and some properties of the set with positive concentration

existence and regularity of extrema abstract consider the minimization of a possibly noncoercive g teaux differentiable functional f x r a modified notion of coercivity is introduced which may be usable to show existence of a minimum alternatively f d r has a minimum at y e d f not differentiable but the restriction f of f to x d differentiable one may be able to show y is actually in x the latter case is related to justification of formally calculated necessary conditions for optimal controls the arguments are applications of ekeland s approximate variational principle j math anal appl 47 1974 324 353

global gradient estimates for non quadratic vector valued parabolic quasi minimizers abstract we establish local and global higher integrability estimates for the gradient of vector valued parabolic quasi minimizers of integral functionals with polynomial p growth p 2 n n 2 on a parabolic domain t in r n 1 n 2 and under a weak regularity assumption on the parabolic boundary of t

classification of blowup solutions for a parabolic p laplacian equation with nonlinear gradient terms abstract we mainly consider the following degenerate p 2 parabolic equation u t p u u m u q with homogeneous dirichlet boundary condition in a bounded domain r n before studying the properties of the solutions of this equation we first establish the local in time existence of its weak solutions then in different ranges of reaction exponents we give the complete classification of blowup results including l blowup and gradient blowup moreover under the subcritical condition of q p 1 m p 1 we can also prove that the solution is global in time

continuity estimates for porous medium type equations with measure data abstract we consider parabolic equations of porous medium type of the form u t div a x t u d u in e t in some space time cylinder e t the most prominent example covered by our assumptions is the classical porous medium equation u t u m in e t with m 1 we establish a sufficient condition for the continuity of u in terms of a natural riesz potential of the right hand side measure as an application we come up with a borderline condition ensuring the continuity of u more precisely if l n 2 2 1 then u is continuous in e t