No Local L 1Solution for a Nonlinear Heat Equation
No Local L 1Solution for a Nonlinear Heat Equation
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非线性热方程无局部 L 1 解
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
Zhengfang Zhou
中科院分区：
文献类型：
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作者：
C. Çelik;Zhengfang Zhou
本文考虑具有某些边界条件和初值条件的非线性热方程，其中p>；1.对于一维情形，众所周知，对于p<；3，这个问题在任何初值条件下都有局部解.然而，L 1关于临界指数p=3的局部解的存在唯一性一直是公开的，本文就是为了回答这个问题。首先证明了L[1]中的柯西问题对于某个u0∈L[1]不存在局部解，然后用截断函数的方法证明了柯西问题解的非局部存在性，从而证明了回答这一公开问题的Dirichlet问题解的非局部存在性.此外，我们还推广了具有临界指数的n维情形的非局部存在性结果。对于Dirichlet边值问题，我们也考虑了更一般的非线性。最后，我们证明了具有相同初值u0的混合边界条件的相同结果。
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.