Entire solutions with moving singularities for a semilinear heat equation with a critical exponent
Tóm tắt
We consider the semilinear heat equation
$$\partial _t u -\Delta u=u^{N/(N-2)}$$
in
$$\Omega $$
with
$$u=0$$
on
$$\partial \Omega $$
, where
$$N\ge 3$$
and
$$\Omega $$
is a smooth bounded domain in
$$\mathbf {R}^N$$
. Let
$$\xi :\mathbf {R}\rightarrow \Omega $$
satisfy
$$\overline{\{\xi (t);t\in \mathbf {R}\}}\subset \Omega $$
. Under some assumption on the uniform Hölder continuity of
$$\xi $$
, we construct a nonnegative solution u defined for all
$$t\in \mathbf {R}$$
satisfying
$$u(x,t)\rightarrow \infty $$
for each
$$t\in \mathbf {R}$$
as
$$x\rightarrow \xi (t)$$
.
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