Study Of Some(Pseudo) Parabolic Equations With Sources And P(x)-Laplace Operator

In this dissertation,the properties of solutions for some(pseudo)parabolic equations with sources and p(x)-Laplace operator are studied.We mainly investigate the effect of the nonlocal sources,the logarithmic nonlinearity source and variable exponents on the existence and blow-up of solutions.This dissertation is divided into four chapters.In Chapter 1,we firstly introduce the background of the problems and the related works in China and abroad,and then we state the problem studied in this dissertation and some methods and techniques used.In Chapter 2,we investigate the initial-boundary value problem for p-Laplace equation with the nonlocal source(?) here T>0,Ω is a bounded domain in RN(V≥1),(?)Ω,is the smooth boundary,n is the unit outward normal on(?)Ω.The initial value u0(x)∈L∞ ∩Wl,p(Ω),uo(x)≠0,∫Ω u0(x)dx=0.The exponents p and q are real numbers satisfying p>2,q>p-1.In 2013,Qu and Liang[1]studied Problem(4).By combining Sobolev embedding Wl,P(Ω)→Lq+1(Ω),they proved that if p-1<q≤Np/(N-p)+-1 and the initial energy satisfies a suitable condition,the solution of Problem(4)blows up in finite time.The difficulty is that the Sobolev embedding W1,p(Ω)→Lq+1(Ω)no longer holds for q>Np/N-p-1(p<N),the methods used in[1]are no longer effective.We construct a new control function,and then combine the concavity method proposed by Levine and Payne[2,3]to prove that the solution of Problem(4)blows up in finite time for q>p-1>1 and suitable initial energy.In Chapter 3,we study the initial-boundary value problem for the following fourth-order equation with the logarithmic nonlinearity and p-Laplace operator(?) here T>0,Ω is a bounded domain in RN(N≥1),(?)Ω,is the smooth boundary,n is the unit outward normal on(?)Ω.The initial value u0∈H02(Ω),and the exponent p satisfies 2<p<∞,N=1,2;2<p<2N/N-2,N≥3.Generally speaking,we do not admit the usual maximum principle and comparison principle for the higher-order equations,which makes some most effective methods,such as the method of upper and lower solutions,invalid any more.In addition,we cannot directly deduce from ▽Vn(?)▽u weakly to div(|▽un|p-2▽un)(?)div(|▽u|p-2▽u)weakly.And the presence of the logarithmic nonlinearity causes some difficulties in deploying the potential well method.To some extent,these three problems cause some difficulties to study Problem(5).In this chapter,to begin with,for the subcritical and critical initial energy,we combine the Galerkin’s method,the logarithmic Sobolev inequality,the modified potential well method and the concavity method to investigate the global existence,blow-up and extinction of the solution for Problem(5).Secondly,for supercritical initial energy,we give some sufficient conditions for the existence of global and blow-up solutions to Problem(5)based on the idea that comes from Gazzola and Weth[4].Next,we give some new blow-up conditions(do not depend on the depth of the potential well),and obtain the upper and lower bounds of the blow-up time.In Chapter 4,we consider the initial-boundary value problem for the following pseudo-parabolic equation with variable exponents and p(x)-Laplace operator(?) here T>0,Ω is a bounded domain in RN(N>1),(?)Ω,is the smooth boundary.The initial value uo∈W01,p(x)(Ω)∩H01(Ω).The exponents p(x)and q(x)are log-Holder continuous functions in Ω,i.e.p(x)and q(x)are measurable functions satisfying(?)(?) and|p(x)-p(y)|+|q(x)-q(y)|≤c/og(e+1/|x-y\)’for any two disjoint points x,y∈Q,here c>0 is constant.As far as we know,there are few works concerning about equations with pseudo-parabolic viscous Aut and variable sources.Particularly,the existence of variable exponents will cause many difficulties.In 2017,Problem(6)was first studied by Di,Shang and Peng[5].They proved that if the initial energy is non-positive,the solution blows up in finite time by means of a differential inequality technique,meanwhile they gave the upper and lower bounds of blow-up time.However,they did not discuss the other properties for Problem(6).In this chapter,firstly,we prove that when the initial energy is bounded,the solution of Problem(6)blows up in finite time by using the energy functional and embedding theorem;we discuss the blow-up problem for arbitrarily high initial energy by defining some functionals and sets,and constructing the ω-limit set;using an extended form of the concavity method obtained by Khelghati and Baghaei[6]and a differential inequality technique,when there does not exist a upper bound for exponent q(x),blowing up in finite time for the solution of Problem(6)is proved.Secondly,for p-≥2,we establish the qualitative relations between the energy functional and source term,and then exploit a key integral inequality to prove the asymptotic stability for solutions of Problem(6).Finally,for q+<2,the global existence and uniqueness of weak solution for Problem(6)are obtained by using Galerkin’s method、Aubin-Lions-Simon compactness lemma and the monotone operator theory.Further,the extinction and non-extinction of the weak solution are discussed.