No Local <i>L</i> ¹ Solution for a Nonlinear Heat Equation

Abstract

Abstract In this article, we consider the nonlinear heat equation on with some boundary conditions and the initial condition , where and p > 1. For the one dimensional case, it is well known that for p < 3 this problem has a local solution for any initial condition . But the existence and uniqueness of a local solution in L 1for the critical exponent p= 3 was wide open and this work is to answer this open question. First, we prove that for the Cauchy problem there is no local solution in L 1for some u 0∈ L 1. Then using the nonlocal existence of Cauchy problem by a cutoff function argument, we prove the nonlocal existence of a solution for the Dirichlet problem which answers this open question. Moreover, we generalize the nonlocal existence result for n-dimensional case with the critical exponent . More general nonlinearity is also considered for Dirichlet boundary value problems. Finally, we prove the same result for the mixed boundary condition with the same initial data u 0.