Fujita blow-up phenomena of solutions for a Dirichlet problem of
parabolic equations with space-time coefficients
Abstract
This paper deals with a homogeneous Dirichlet initial-boundary problem
of parabolic equations with different space-time coefficients, $$u_t
=\Delta u + t^{\sigma_1}
u^{\alpha} + \langle
x\rangle^{n} v^{p},\quad v_t
=\Delta v + \langle
x\rangle^{m} u^{q} +
t^{\sigma_2} v^{\beta},$$
where the eight exponents are nonnegative constants and
$\langle x\rangle$ is the Japanese
brackets. We obtain the Fujita exponents of solutions, which are
determined by the eight exponents and the dimension of the space domain.
Moreover, simultaneous or non-simultaneous blow-up of the two components
of blow-up solutions is discussed with or without conditions on the
initial data.