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

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

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

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

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

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

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

revisit on partial solutions in the adomian decomposition method solving heat and wave equations the adomian decomposition method is considered in application to heat and wave equations the so called partial solution technique is used it is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions in cases involving inhomogeneous boundary conditions improper results may be obtained by former method this paper presents a further insight into partial solutions in the decomposition method and the resolution of such cases

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

asymptotic behavior of positive solutions of a nonlinear integral system abstract in this paper we study the properties of positive solutions of an integral system in r n u x r n v p y u q y x y n y d y v x r n v q y u p y x y n y d y we prove the radial symmetry of positive solutions by means of the method of moving planes in integral forms which is established by chen li ou in addition by using a regularity lifting lemma we obtain an integrability result under the assumption p q 1 0 we prove that those solutions are the ground states finally we estimate the decay rates of such solutions when x

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

experimentation with two step adomian decomposition method to solve evolution models the two step adomian decomposition method tsadm is derived by separate the standard adomian decomposition method into two steps although the tsadm needs only a slight variation from the standard adomian decomposition method experimentation shows that the tsadm is also promising for systems of inhomogeneous differential and integral equations hyperbolic partial differential equations and singular initial value problems

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

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

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

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

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

a new computational method for laplace transforms by decomposition method in this paper adomian decomposition method is applied to develop a new computational method for laplace transforms some examples are provided to illustrate the simplicity and reliability of the new method

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

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

positive solutions to integral systems with weight and bessel potentials abstract in this paper we consider the integral system with weight and the bessel potentials u x r n g x y u y p v y q y d y v x r n g x y v y p u y q y d y where u v 0 0 0 n p q 2 and g x is the bessel potential of order first we get the integrability by regularity lifting lemma then we also establish the regularity of the positive solutions afterwards by the method of moving planes in integral forms we show that the positive solutions are radially symmetric and monotone decreasing about the origin finally by an extension of the idea of lei 14 and analytical techniques we get the decay rates of solutions when x

decay symmetry and existence of solutions of semilinear elliptic systems our utmost aim was originally to establish the existence of solutions of system however in this effort we got sidetracked to consider other questions in this way we come to interesting results on asymptotic behavior of solutions on symmetry properties of such solutions and the existence of ground states

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

a distinctive sumudu treatment of trigonometric functions abstract the sumudu transform integral equation is solved by continuous integration by parts to obtain its definition for trigonometric functions the transform variable u is included as a factor in the argument of f t and summing the integrated coefficients evaluated at zero yields the image of trigonometric functions the obtained result is inverted to show the expansion of trigonometric functions as an infinite series maple graphs tables of extended sumudu properties and infinite series expansions of trigonometric functions sumudi images are given

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

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

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

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

regularity of solutions of nonlinear degenerate parabolic systems soit le systeme parabolique u i t div u p 2 u i 0 i m avec p 1 ou u i u i x t grad x et x r n ouvert pour toute solution de ce systeme avec max 1 2n n 2 p u est continu dans t

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

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