Author:
Chung Soon-Yeong,Hwang Jaeho
Abstract
AbstractThe purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations $$ u_{t}=\Delta u+\psi (t)f(u),\quad \text{in }\Omega \times (0,\infty ), $$
u
t
=
Δ
u
+
ψ
(
t
)
f
(
u
)
,
in
Ω
×
(
0
,
∞
)
,
under the mixed boundary condition on a bounded domain Ω. In fact, this has remained an open problem for a few decades, even for the case $f(u)=u^{p}$
f
(
u
)
=
u
p
. As a matter of fact, we prove: $$ \begin{aligned} & \text{there is no global solution for any initial data if and only if } \\ & \int _{0}^{\infty}\psi (t) \frac{f (\lVert S(t)u_{0}\rVert _{\infty} )}{\lVert S(t)u_{0}\rVert _{\infty}}\,dt= \infty \\ &\text{for every nonnegative nontrivial initial data } u_{0}\in C_{0}( \Omega ). \end{aligned} $$
there is no global solution for any initial data if and only if
∫
0
∞
ψ
(
t
)
f
(
∥
S
(
t
)
u
0
∥
∞
)
∥
S
(
t
)
u
0
∥
∞
d
t
=
∞
for every nonnegative nontrivial initial data
u
0
∈
C
0
(
Ω
)
.
Here, $(S(t))_{t\geq 0}$
(
S
(
t
)
)
t
≥
0
is the heat semigroup with the mixed boundary condition.
Funder
National Research Foundation of Korea
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
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