No Local L¹Solution for a Nonlinear Heat Equation

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.

Keywords:

• Local solution
• Nonlinear heat equation
• Critical exponent
• AMS Classification:35K55
• 35K57
• 35K05
• 35B33

Acknowledgments

We would like to thank the referee of this article for corrections and helpful suggestions that improved some of our arguments. Also we would like to thank Prof. Clifford Weil for his corrections for typing this article.