| In the recent years, many mathematicians are interested in studying partial differen-tial equations with variable exponents, for some parts of the work, the interested readers may refer to [44] and the references therein. The main reason lies in its vital and impor-tant applications in physics and in the real world. The models governed by PDEs with variable exponents are mainly from electro-rheological fluids model [99]. It provides more precise and appropriate explanations for mechanical properties of some electro-rheological fluids with viscosity. The PDEs models with variable exponents describe a physical phe-nomenon for conductor, which characterized by their ability to undergo significant changes in their mechanical properties when an electric filed exerted from outside. Without the electro-rheological fluids model and their properties, in some sense it would not be the appearance of medical rehabilitation equipment, shock wave absorber, actuator, clutch etc.The PDEs with variable exponents also describe some evolutional phenomenon in thermodynamics, heat and mass transfer in nonhomogeneous media and non-Newtonian fluids with thermo-convective effects [9]. There are other applications in the calculus of variations [35], elasticity [116], image denoise and image restoration [34], etc. In particu-lar, in the digital image processing, the consideration of nonstandard growth condition has more advantages, chief among which is the so called staircase effect. Or more precisely, the study of variational integrals with nonstandard growth condition not only preserves edges in the picture, but also creates new edges where there were none in the original image. By virtue of the 'staircase effect', it helps to carry out the image restoration.This thesis mainly discuss the existence of weak solutions, renormalized solutions and entropy solutions to some parabolic and elliptic equations with variable exponents. It should be remarked that the partial differential equations concerned are investigated in the framework of variable exponent Sobolev space. The main interest includes the weak solution to doubly degenerate parabolic equation with nonlocal term; the weak solution and renormalized solution to nonlinear parabolic p(x)-Laplace equation with natural growth condition; the renormalized solution and entropy solution to an elliptic p(x)-Laplace equation with zero order term and degenerate coercivity.Chapter 1 is the introduction to the main content of this thesis as well as some pre-liminary results on the Lebesgue and Sobolev spaces with variable exponents. Comparing with the constant exponents case, that the difficulties we encounter in the framework of the variable exponents is our core in this part. We also tell the main methods and tools to our problem.In chapter 2, we study the following parabolic p(x)-Laplace equation: Here, ? is a bounded and simply connected domain in RN with smooth boundary (?)?. QT= ?× (0,T), ?T= (?)?× (0,T), T> 0, ? ? (0,+?). We assume that m> 1, p ? C1,?(?); and p+:= max p(x), p-:= min p(x), p+?p-? 2. The functions a ? L?(QT), K ? L?(QT) can be extended to ?×R by T-periodicity. Furthermore, there holds K ? 0 a.e. (x,t) ? QT.With the aid of Leray-Schauder topological degree, we obtain the existence of non-negative and nontrivial weak solution to Problem (2.1). First of all, we construct a regularized equation to Problem (2.1). The next work is to give the upper bound and lower bound estimates to the solution sequence of the regularized problem. The critical step is the estimation of the bound from above. Employing the modified De Giorgi iter-ation technique, we get the uniform bound from above. By picking up appropriate test function, we have the bound from below. Thanks to the homology invariance of Leray-Schauder topological degree, we find the weak solution to the regularized equation in an annulus. At the end, we enjoy the nontrivial and normegative weak solution to Problem (2.1) by Minty's argument and a limit process.In chapter 3, we consider the following initial and boundary problem with respect to a parabolic p(x)-Laplace equation with gradient term: Here, ??RN N is a bounded domain with smooth boundary (?)?, QT=?× (0,T), ?T= (?)?×(0,T), T> 0 is finite, p(x), B (x,t), F(x,t) are given quantities. B ? L?(QT) satisfies 0 ? B(x,t) ? b, where b > 0 is a constant. F is a vector field with |F|(p-)'?L*(QT),where (p-)'=p-/(p-' -1) and N+p-/p-By virtue of the method of L? estimate, we investigate the existence of weak solu-tions in the space V defined in chapter 3 to Problem (3.1). First, in order to get the L? bound, we have to modify the De Giorgi iteration technique since the problem we study is in the setting of variable exponents. Once the uniform L? bound for Problem (3.1) get, we can pay attention to its perturbed problem, which is generated by approximating the gradient term on the right-hand side. Utilizing the similar De Giorgi iteration to the perturbed problem, one is able to get the L? bound for the approximating solution sequences un. We employ the method of nonlinear test functions in [53] to the approxi-mating problem and have the a priori estimates. In the end, through a limit process we obtain the existence of weak solutions to Problem (3.1).The previous two chapters concern with the weak solutions to parabolic equations with nonstandard growth conditions (one main reason is that the terms on the right-hand side of the equation have good integrability). In chapter 4 and chapter 5, we concentrate on the renormalized solutions and entropy solutions of parabolic and elliptic equations with lower order terms and variable exponents.Chapter 4 focus on the following nonlinear parabolic equation with gradient term and nonstandard growth condition: Here, ??RN is a bounded domain with smooth boundary (?)?,QT= ?× (0,T), ?T= (?)?× (0, T), T> 0 is finite;f?L1(QT), u0?L1(?).F is a vector function satisfies F?(Lp'(x)(QT))N, where 1/p(x)+1/p'(x)= 1. Functio g(x,t,s,?):QT×R×RN?R satisfies Caratheodory condition and there holds |g(x,t, s,?)|?h(|s|)|?|p(x)+?(x,t), where h is a positive and nondecreasing continuous function, ?> 0 and ??L1(QT). Moreover, g satisfies the following sign condition g(x,t,s,?)s?0, for any (s,?)?R×RN and almost every (x,t) ? QT.After dealing with the initial value u0, gradient term g(x, t, u, ?u) and right-hand side term f by truncation approximation, we have the approximate equation corresponding to Problem (4.1). The approximating solution sequence is also denoted by un. Recalling the method used in the previous chapter 3 to get the almost everywhere convergence of un. In fact, that the uniform L? bound obtained through De Giorgi iteration plays an essential role. With the help of L? bound, we have that un is bounded in the weak solution space V and (?)up/(?)t is bounded in V*+L1(Qt), then it follows that the strong convergence of un and almost everywhere convergence of un. Nevertheless, in chapter 4, the integrability of the right-hand side term in Equation (4.1) is not good enough, which tells us that the method used in chapter 3 is in vain for chapter 4. We adopt the method of truncation instead. First, through some a priori estimates for Tk(un) defined in chapter 4, we prove that un converges in measure. Then, based on Riesz theorem, we extract the subsequence of un and obtain its almost everywhere convergence. Next, we claim that the critical work is to take appropriate test functions to the approximate equation and make the needed estimates, which is rather technical and complexity. In the end, proceeding a limit process, we embrace the existence of renormalized solutions to Problem (4.1).In the setting of variable exponent Sobolev space, we, in chapter 5, consider the renormalized solutions and entropy solutions to the following nonlinear elliptic equation with zero order term and degenerate coercivity Here, ??RN is a bounded domain with smooth boundary (?)?; f ? L1(?). a(x,s) ?×R?R is a Caratheodory function satisfying where ? > 0,?(x) ? C(?), ?(x)?0 (in fact, this condition shows that the principal term -div[a(x,u)|?u|p(x)-2?u] in equation (5.1) has degenerate cocivity); Moreover, where ??[0,+?) ? (0,+oo) is a continuous function.Function g(x, s):?×R?R satisfies Caratheodory condition, and for any k ?R+, there holds Furthermore, g satisfies the following sign condition g(x, s)s ? 0, for any s ? R, for almost every x ? ?In this chapter, truncation method is also our main tool to have the a priori estimates for the approximate equation (again we denote its weak solution sequence by un). Howev-er, there are many differences comparing with what we have done in chapter 4. Recalling chapter 4, in the parabolic cases, utilizing the information on the initial value uo and taking appropriate test function, we can obtain that un is bounded in L?(0,T; L1(?)); and it is bounded in L1(QT)-But this method is not suitable to elliptic equations if one wants to get that un is bounded in L1(?). In order to conquer this difficulty, we seek the Marcinkiewicz estimate with variable exponent [100J for help. The application of Marcinkiewicz estimate with variable exponent has many advantages. On one hand, it surpasses the difficulty from the degenerate coercivity term. On the other hand, it helps us to obtain the information of the regularity on u. Moreover, with the help of Marcinkiewicz estimate with variable exponent, not only can we obtain the almost ev-erywhere convergence of approximating solution sequence un, but also can deal with the zero order term. By choosing appropriate test functions and a limit process, we have the strong convergence of Tk(un) in W01,p(x)(?), based upon which we prove the existence of renormalized solutions and entropy solutions to Problem (5.1). |
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