PARABOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS WITH LOCALIZED NONLINEARITY

Abstract

Behavior of solutions of nonlinear partial differential equations is delicately influenced by the nonlinear terms. Such phenomenon is of interest in the theory of partial differential equations, in the context of mathematical modeling, as well as in the area of mathematical physics. In this paper, particularly interested in the effect of a compactly supported coefficient added on the nonlinear terms, we study the behavior of positive solutions to porous medium equation with localized reaction, and semilinear elliptic equation with localized nonlinearity. The first half mainly deals with the critical exponents concerning the largetime behavior of positive solutions to porous medium equation with localized reaction in multi-dimensional space. We concluded our results with two main theorems – the two-dimensional case and the higher dimensional case. Especially for the latter one, namely, when the space dimension is not less than three, we clarified the relationship between the behavior of nonnegative solutions and the exponents contained by the diffusion and reaction terms of the equation. In addition, for further discussion of the support of blow-up solutions, a property concerning the support of solutions is also proved. In the second half, to continue to study the effect of the localized nonlinearity on the behavior of solutions to partial differential equations, we studied the role of a localized coefficient in a priori estimate for positive solutions to the semilinear elliptic equation. For the semilinear elliptic equation without localized nonlinearity, the existence of an a priori bound for all positive solutions is a wellknown result. However, we discovered that under the influence of the localized nonlinearity, certain conditions should be imposed to guarantee the existence of the a priori bound. In our two main theorems, we respectively obtained two types of such conditions for the existence of the a priori bound. Furthermore, for future work, we suggested possible improvement of the result, and presented a corresponding semilinear parabolic problem where our arguments and techniques may be applicable.