On linear and nonlinear heat equations in degenerating domains

Abstract

Earlier we studied the homogeneous boundary value problem for the heat equation in degenerating domains. For this problem in the weight class of essentially bounded functions it was established the existence of a nontrivial solution up to a constant multiplier. In this paper, on the basis of the above result, we study the issues of the existence of nontrivial solutions of homogeneous nonlinear heat equations, including the homogeneous Burgers equation in degenerating domains. The nonhomogeneous boundary value problems for the Burgers equation are studied separately.