Study Of Solutions To Some Nonlinear Parabolic Equations With Nonstandard Growth Conditions

This paper deals with the properties of weak solutions to classes ofnonlinear parabolic equations with nonstandard growth. These problemsarise from modeling physical problems, such as electro-rheological fluids,nonlinear elastic mechanics, image processing etc. To the best of our knowl-edge, there are many works about elliptic equations with variable exponentsof nonlinearity, but there are only a few works about parabolic equationswith variable exponents of nonlinearity. This paper is divided into fourchapters:In Chapter one, we state some history about nonlinear parabolic equa-tions, and variable exponent Sobolev spaces. At the same time, we givesome examples to illustrate the diference between classical Sobolev spacesand variable exponent Sobolev spaces. At the end of the chapter, we simplygive our problems which we will discuss and some methods and techniqueswhich we will apply.Next, Chapter two will be devoted to the study of nonlinear parabolic problem with p(x, t) satisfyingwhereSince the function a(u) in div(a(u)|u|p(x,t)2u) of the above prob-lem may not have an upper-bounded, this brings us some difculties. Inorder to overcome these difculties, we apply the method of parabolic reg-ularization and Galerkin's method to prove the existence of weak solutionsto the problems mentioned. Furthermore, we obtain the uniqueness ofweak solutions with an argument of contradiction. Our main results aboutthe existence of weak solutions areTheorem1Let p(x, t) satisfy Conditions (2)(3). If the followingconditions holdThen Problem (1) has at least one weak solution. According to the relationship between σand p±,we have the following results about the uniqueness of weak solutionsTheorem2Suppose that the conditions in Theorem1are fulfilled is unique within the class of all onn-negative weak solutions. Theorem3Suppose that the conditions in Theorem1are fulfilledand the following condition is satisfied Then the nonnegative bounded solution of Problem(1)is unique within the calss of all nonnegative bounded weak solutions. Moreover,we construct a suitable test-function and apply some skills in convex analysis to obtain the localization property and the long time asymptotic behavior of weak solutions Theorem4Assume that the conditions of Theorem1are fulfilled and(1)and f≡0,then. bounded nonnegative solution of Prolem(1)vanishes in finite time for any nonnegative initial data0≠u0∈L∞(Ω)∩W1,p(x)(Ω)and satisfies thefollowing estimates where C1is a positive constant.finite time,for any nonnegative initial data0≠u0∈L∞(Ω)∩W1,p(x)(Ω) and satiSfies the following estimatesTheorem7Suppose that f(x)三0,2≤p-<p+,If there exists a Then for all0＜σ,the solution to problem (1) satisfiesIn Chapter three.we discuss the properties of weak solutions of a class of the initial Neumann problem with degeneracy or singularity Since the functioil spaces to which the solution belong in Chapter one are not able for our discussing Neumann problem,we have to find a suitable function space to study the problem mentioned.Fortunately.we find out a function space which is suitable for our study.at the same time.we also prove some properties about functions in the space,such as separabil-ity.reflexivity.completeness and so on as well as the densitv of a class of infinite differentiable functions in our function spaces constructed,which will provide some theoretical bases for applying Galerkin's approximiation technique. And then we transform our problem into ODE'problems by Galerkin's approximation technique,and obtain the existence of solutions according to the classical results of O DE.Furthermore,making necessary uniform estimates and a compactness argument,we prove that the limit function of approximate solutions is a weak solution to the problem men-tioned.First,we consider the case f(x,t,u)≡0.Our main result isTheorem8Let p(x,t)satisfy Conditions(2)-(3).If the following conditions hold then there exists a positive congstant T0(q±,||u0||∞Ω,|Ω|)such that Problem (4)has a uniguc solution u(x,t)∈W(QT0)∩L∞(0,To；L*2(Ω)).Besides,we apply energy estimate method and the comparison prin-ciple for ODE to prove the properties of extinction in finite time to the weak solution.First,we consider the case F(x,t)=0.Theorem9Let u(x,t)∈W(QT)QL∞(0,T；L*2(Ω))be a weak solution of Problem(4)；If the following conditions are satiisfied then the solution of Problem(4)vanishes in finite time T1*with T1*satisfies Secondly,we discuss the case divF≠0,we haveTheorem10Let¨∈W(QT)∩L∞(0,T；L*2(Ω))be a weak solution of Problem(4).If the following conditions hold(H11)there exists a positive constant T0*such that then the solution of Problem(4)vanishes in finite time T2*satisfying0T2*≤T0*. Secondly,we consider the case div户三0,f(x,t,u¨)=|u|r-2-|Ω|-1∫Ω|u|r-2udx Our main results areTheorem11Assume that p(x,t)satisfies conditions(2)-(3)and the following conditions hold then Problem(4)has at least one solution. SetLet B be the optiimal constant in the following imbedding that is Define where α satisfies Our result isTheorem12Suppose that p(x,t)≡p(x)satisfies(2)-(3)and the following conditions are satisfied then the solution of problem(4)blows up in finite time.In final chapter,we consider a class of mathematical models arising from image processing-higher-order nonlinear parabolic problem involving p(x)-Laplace operatorFirst,since the general maximum principle for second-order equations fails in higher-order parabolic equations,we can't generalize the lower-upper solutions technique to the case of higher-order parabolic equations. Secondly,the variable exponent p(x)produces some diffculties in mak-ing a prior estimates. Besides,the nonlinear source f(x,u))will certainly bring some difficulties in the proof of the existence,for instance,since it is nonlinear,a large number of techniques(such as variational method)which can be applied in the linear cases(when p(x)is a positive constant)fail. In this paper,we combine the differential and variational techniques to obtain the existence of weak solutions to the fourth-order elliptic problem with a linear source f(x)and prove the existence of weak solutions of the steady state problem of Problem(5)by applying Leray-Schauder's fixed point theorem. Moreover,we apply the time-discrete technique to prove the existence of weak solutions of the evolution problems with necessary uniform estimates and a compactness argument. Unlike the previous pa-pers,we apply the iteration technique toobtain the higher integrability of the time derivative. At first,we consider the steady-state problem of Problem (5),this isWe apply variational method and Leray-Schauder fixed point theo-rem to obtain the existence of weak solutions to the above problem with necessary uniform estimatesTheorem13Let u0∈W01,P(X)(Ω).If the exponents p(x),m(x) and the nonlinear source f(x,u¨)satisfy Conditions(2)-(3)and Then Problem(6)has at least one weak solution.Furthermore we obtain the uniqueness of weak solutionsTheorem14Suppose that f(x,u)=-k(u(x)-a(x))with k>0and p->2,then Problem(6)admits at most one weak solution.Next,we apply the time-discrete technique and make a priori estimates to prove the existence of weak solutions to Problem(5)Theorem15Suppose that the exponent p(x)satisfies Conditions(2)一(3)and p->2.If the following conditions hold (H18)u0∈H01(Ω)∩W03,p(x)(Ω),f(x)∈H01(Ω)；(H19)|div(|(?)u0|p(x)2(?)△0)|∈H01(Ω), then Problem(5)has a unique weak solution. Finally,we obtain the regularity of weak solutions by applying a modified techniqueTheorem16Suppose that the exponent p(x)and the function f(x) satisfy the conditions in Theorem15.Let u(x,t)be a weak solution of Problem(5)with N=1,Ω=(0,1).Then U∈C2α,α(QT)with α=1/p-...