an oscillating solution for the porous medium equation with initial condition in l r which did not stabilize at x 0 this indicated that the mechanism of diffusion in the equation of heat conduction is not sufficiently strong to smooth out highly nonuniform albeit bounded distributions of initial data later on repnikov and eidelman 20 21 gave necessary and sufficient conditions for stabilization of solutions of certain parabolic equations including 1 3 their result stated that within the class of uniformly bounded solutions one has

global existence and asymptotic behavior for some multidimensional quasilinear hyperbolic systems abstract for the cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self interaction we show the global existence of classical solutions with small initial data we also prove that the global solution will converge to a solution of the linearized system as time tends to infinity and the convergence rate is also determined

long time regularity of solutions of the hele shaw problem abstract in this paper we study the long time behavior of solutions of the one phase hele shaw problem without surface tension we show that after a finite time solutions of the hele shaw problem become starshaped and lipschitz continuous in space based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed

weak solutions of the porous medium equation in a cylinder we show that if d r n is a bounded domain with smooth boundary and u l m d e t u 0 solves u t u m m 1 in the sense of distributions on d 0 t and vanishes on d 0 t in a suitable weak sense then u is holder continuous in d 0 t

remarks on energy approach for global existence of some one dimension quasilinear hyperbolic systems abstract we show that luli yang and yu s weighted energy estimate with positive weights 8 together with some variants of alinhac s ghost weight energy approach 1 can be used to prove the global existence of classical solutions to some one dimension quasilinear hyperbolic systems of diagonal form with small initial data

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

localization for the evolution p laplacian equation with strongly nonlinear source term abstract in this paper we study the strict localization for the p laplacian equation with strongly nonlinear source term let u u x t be a solution of the cauchy problem u t div u p 2 u u q u x 0 u 0 x where x t r n 0 t n 1 and p 2 when q p 1 we prove that if the initial data u 0 x has a compact support then the solution u t has also compactly support moreover when 1 q p 1 we show that the solution of the cauchy problem blows up at any point of r n to arbitrary compactly supported initial data

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

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

interfaces with a corner point in one dimensional porous medium flow we prove that for a large class of initial distributions the solutions of the initial value problem for the one dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time this happens for example if the initial pressure is o x 2 as x 0 we also give sufficient conditions for the smoothness of the interface which improve previous results

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

a fatou theorem for the solution of the heat equation at the corner points of a cylinder in this paper the author proves existence and uniqueness of the initial dirichlet problem for the heat equation in a cylindrical domain d 0 where d is a bounded smooth domain in r n with zero lateral values a unique representation of the strong solution is given in terms of measures on d and on d we also show that the strong solution u x t of the heat equation in a cylinder converges a e x 0 d 0 as x t converges to points on d 0 along certain nontangential paths

resonance type behaviour in a system of nonlinear wave equations consider the cauchy problem for a system of two wave equations in r3 c12 t2u1 u1 u 1 u 2c22 t2 u2 u 1 u 2 where prime stands for any partial derivative in space or time variables if initial data is small in a suitable norm the following phenomenon occurs if c1 c2 the system possesses smooth solutions for all t 0 whereas if c1 c2 the solutions of the system might blowup the result is proved for a more general form of the above system in the case of a three space dimension also a similar result is proved for two space dimensions

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

nonnegative solutions of the initial dirichlet problem for generalized porous medium equations in cylinders in this paper we study nonnegative solutions u of a class of nonlinear evolutions au at a u whose best known example is the porous medium equation u u m 1 see 9 for a recent survey on these equations we study our solutions u subject to dirichlet boundary conditions in d x 0 t where d is a bounded smooth domain in rn i e uiadx 0 t 0 the corresponding theory for d rn was studied by the authors in 3 we find that there is one special solution x t the friendly giant which tends to infinity as t 1 0 and that all other solutions u x t are in one to one correspondence with suitable pairs of measures u on d and a ou ad we also show that any solution on d x 0 t extends to a solution in d x 0 oo and that the friendly giant f x t governs the asymptotic behavior as t t oo the nonlinearity is assumed to be continuous increasing with p 0 0 and also verifies the growth conditions

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension abstract a weighted energy estimate with tangential derivatives on the light cone is applied for the cauchy problem of semilinear wave equations with the null conditions in one space dimension the well posedness and lifespan of the solutions are considered based on the vector field method

on one dimension semi linear wave equations with null conditions abstract it is well known that in space dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data it is also known that in space dimension two such result still holds for a certain class of quasi linear wave equations with null conditions the proofs are based on the decay mechanism of linear waves however in one space dimension waves do not decay nevertheless we will prove that small data still lead to global solutions if the null condition is satisfied

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

feynman kac functionals and positive solutions of 1 2δu qu 0 let d be a bounded c 2 domain in d and let q be a bounded borel function in d for x d and z d suppose x t under the law p x z is brownian motion in d starting at x and conditioned to converge to z let be the lifetime of x t we show that if the quantity e x z left exp left int limits 0 t q x s ds right right is finite for one x d and one z d then this quantity remains bounded as x varies over d and z varies over d this may be considered one quantitative expression of the qualitative statement that no matter where brownian motion in d eventually hits d it goes all over d before it gets there we apply this result to show that if the equation 1 2 u qu 0 admits a non negative solution in d which is strictly positive on a subset of d of positive harmonic measure then for any non negative bounded borel function f on d it admits a unique bounded solution u satisfying u f on d and this solution u is non negative

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

nonradial solvability structure of super diffusive nonlinear parabolic equations we study the solvability of the cauchy problem for the nonlinear parabolic equation u div u m 1 u at when m 0 then there is no local solution of the initial value problem on the other hand if such an interval does not exist then the initial value problem is locally solvable and the time of existence can be estimated explicitly

a remark on infinite initial values for quasilinear parabolic equations abstract we study the possibility of prescribing infinite initial values for solutions of the evolutionary p laplace equation in the fast diffusion case p 2 this expository note has been extracted from our previous work when infinite values are prescribed on the whole initial surface such solutions can exist only if the domain is a space time cylinder

upper bounds on waiting times for the thin film equation the case of weak slippage we derive upper bounds on the waiting time of solutions to the thin film equation in the regime of weak slippage n in 2 frac 32 11 in particular we give sufficient conditions on the initial data for instantaneous forward motion of the free boundary for n in 2 frac 32 11 our estimates are sharp for n 2 they are sharp up to a logarithmic correction term note that the case n 2 corresponds with a grain of salt to the assumption of the navier slip condition at the fluid solid interface we also obtain results in the regime of strong slippage n in 1 2 however in this regime we expect them not to be optimal our method is based on weighted backward entropy estimates hardy s inequality and singular weight functions we deduce a differential inequality which would enforce blowup of the weighted entropy if the contact line were to remain stationary for too long

a limit case in nonlinear diffusion on considere le probleme suivant u t u 2 u x x si t 0 x r u 0 x u 0 x si x r cette equation est un modele de diffusion rapide en conduction de chaleur non lineaire on suppose que u 0 x est une fonction reelle localement integrable et non negative on determine des conditions sur u 0 qui assurent l existence d une solution u t x 0 dans un intervalle de temps

universal decay in nonlinear parabolic equations on considere le probleme aux valeurs initiales pour une equation d evolution du dt a u 0 t 0 u 0 u 0 on deduit des taux de decroissance universel pour des problemes d evolution de la forme u t u f u x t 0 u x t 0 x t 0 u x 0 u 0 x x ou r n est borne et et f sont des non linearites satisfaisant certaines conditions de structure

infinitely many homoclinic solutions for a nonperiodic fourth order differential equation without ar condition in this paper we study the existence of infinitely homoclinic solutions for a nonperiodic fourth order differential equation under the superlinear condition but not assuming the well known ar condition the proof is based on variational methods

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

the global stability of minkowski space time in harmonic gauge we give a new proof of the global stability of minkowski space originally established in the vacuum case by christodoulou and klainerman the new approach which relies on the classical harmonic gauge shows that the einstein vacuum and the einstein scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global causally geodesically complete solutions asymptotically convergent to the minkowski space time

homoclinic solutions for a nonperiodic fourth order differential equations without coercive conditions in this paper we investigate the existence of homoclinic solutions for the following fourth order nonautonomous differential equations u 4 wu a x u f x u fde wherew is a constant a c r r and f c r r r the novelty of this paper is that when fde is nonperiodic i e a and f are nonperiodic in x and assuming that a does not fulfil the coercive conditions and f satisfies some more general ar condition we establish one new criterion to guarantee that fde has at least one nontrivial homoclinic solution via using the mountain pass theorem recent results in the literature are generalized and significantly improved

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

singular parabolic equations with measures as initial data abstract in this paper we study the cauchy problem for the singular evolution p laplacian equations with gradient term and source on the assumption of measures as initial conditions for the supercritical case q p 1 p n we obtain that for every nonnegative solution there exists a nonnegative radon measure as initial trace and has some local regularity

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

two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation abstract in this paper we study multiple homoclinic solutions for a class of fourth order differential equations with a perturbation by establishing a compactness lemma and using variational methods the existence result of two homoclinic solutions is obtained under some suitable assumptions but not requiring the periodicity condition some recent results are improved and extended