Linear and nonlinear parabolic partial differential equations with singular lower order term.

摘要

This dissertation gives a treatment of three topics in partial differential equations.; We study the linear heat equation with a highly singular, oscillating potential. We prove that nonexistence of positive solutions is intimately related with the bottom of the spectrum of the stationary problem. Moreover, we prove that all positive solutions blow up instantaneously in some cases, and we characterize these cases. The main feature of the result is the fact that even highly singular, oscillating potentials do not affect how the Baras and Goldstein [4] phenomenon occurs for the heat equation.; We consider the filtration equation and the p-Laplace heat equation with a singular lower order term. Using variational analysis methods, we analyze the nonexistence of positive solutions. We focus on some concrete potentials and determine the critical exponent for the nonexistence of positive solutions.; We study the filtration equation and the p-Laplace heat equation with a singular lower order term on the Heisenberg group. Nonexistence of positive solutions is analyzed. We consider some concrete potentials and determine the critical exponent for the nonexistence of positive solutions.