论文标题
半空间中半线性扩散方程的正溶液的爆炸现象：分散核的影响
Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel
论文作者
论文摘要
我们考虑半线性扩散方程$ \ partial $ t u = au + | u | $α$ u在半空间r n +：= r n -1 x（0， + $ \ infty $）中，其中a是线性扩散操作员，它可能是经典的拉普拉斯操作员，或分数拉普拉斯操作员，或者是适当的非正则非正规化非正规化非局部操作员。该方程是在半空间r n +中无效的初始数据u（0，x）= u 0（x），而dirichlet边界条件u（t，x'，0）= 0，对于x'$ \ in $ r n -1。我们证明，如果操作员A的符号为订单A | $ξ$ | $β$ near the origin $ξ$ = 0, for some $β$ $\in$ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 < $α$ $\le$ $β$/(N + 1). On the other hand, we prove existence of positive global solutions of the semilinear diffusion equation in a half-space when $α$ > $β$/（n + 1）。在半空间的情况下，指数$β$/（n + 1）小于r n中所谓的fujita指数$β$/n。
We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $α$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ' , 0) = 0 for x ' $\in$ R N --1. We prove that if the symbol of the operator A is of order a|$ξ$| $β$ near the origin $ξ$ = 0, for some $β$ $\in$ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 < $α$ $\le$ $β$/(N + 1). On the other hand, we prove existence of positive global solutions of the semilinear diffusion equation in a half-space when $α$ > $β$/(N + 1). Notice that in the case of the half-space, the exponent $β$/(N + 1) is smaller than the so-called Fujita exponent $β$/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).
