Asymptotic Behavior For Some Nonclassical Diffusion Equations

This thesis deals with asymptotic behavior for some evolution nonlinear PDEs, in-volving critical and second critical Fujita exponents, asymptotic profiles for global solu-tions, and life spans for nonglobal solutions. Two of the four nonlinear PDEs considered, i.e., the coupled nonlinear pseudo-parabolic equations and the nonlinear nonlocal diffu-sion equations, belong to nonclassical parabolic equations. We will study the contributions from the high-order viscosity terms in pseudo-parabolic equation and the localization of the source in the nonlocal diffusion problem, respectivcly, to the asymptotic behavior of solutions.The thesis is composed of four chapters:Chapter1introduces the background of the related issues, and briefly summarizes the main results of the present thesis.In Chapter2, we consider the Cauchy problem to a coupled nonlinear pseudo-parabolic system. The local existence of the mild solution is obtained via the fixed-point theorem technique, which implies the existence of the classical solutions under smooth initial data. Furthermore, the comparison principle comes from the positivity of the fun-damental solution associated. Based on these preliminaries, we study the critical Fujita exponent, the second critical exponent, and the asymptotic profile of global solutions. It is observed that the high-order viscosity terms do not change the forms of the two critical exponents and the asymptotic profile. However, the presence of the high order terms bring some substantial difficulties, such as the lost of self-similarity and regularity of solutions, and the complicated form of the fundamental solution involved.In Chapter3, we study the influence of the inhomogeneity of source to the asymptotic behavior of solutions. The first section considers a nonlocal diffusion equation with local-ized source. It is shown that the localization of source significantly affects the asymptotic behavior of solutions, even under nonlocal diffusion. More precisely, there is no Fujita phenomenon if the space dimension N≥2, and the Fujita phenomenon happens only for N=1. In the case of N=1, compared with the situation with local source under nonlo-cal diffusion, both the critical Fujita exponent and the second critical exponent decrease because of the localization factor, which implies that the parameter region for blowing up under any initial data shrinks, and the threshold of initial data for blowing up (in the coexistence parameter region) is enlarged. Next, we consider the Cauchy problem o an evolution p-Laplacian equation with weighted source in the second section, the critical Fujita exponent of which was known. We establish the second critical exponent, as well as the uniform estimate on the life span of non-global solutions for the case of homogeneous source.Chapter4concerns the Fujita phenomenon for the Cauchy problem of an inhomoge-neous fast diffusion system. Both the critical exponent and the second exponent are ob-tained. We observe that the inhomogeneous terms in the system substantially contribute to the critical exponent, in that the blow-up exponent region is obviously enlarged, with keeping the second critical exponent unchanged.