918 research outputs found
Local and global properties of solutions of heat equation with superlinear absorption
We study the limit, when of solutions of
in with initial data k\gd, when is a positive
increasing function. We prove that there exist essentially three types of
possible behaviour according and belong or not to
, where . We emphasize the case where
f(u)=u((\ln u+1))^{\alpha}. We use these results for giving a general result on
the existence of the initial trace and some non-uniqueness results for regular
solutions with unbounded initial data
Initial trace of positive solutions of a class of degenerate heat equation with absorption
We study the initial value problem with unbounded nonnegative functions or
measures for the equation \prt_tu-\Gd_p u+f(u)=0 in \BBR^N\ti(0,\infty)
where , \Gd_p u = \text{div}(\abs {\nabla u}^{p-2} \nabla u) and is
a continuous, nondecreasing nonnegative function such that . In the
case , we provide a sufficient condition on for existence
and uniqueness of the solutions satisfying the initial data k\gd_0 and we
study their limit when according and are
integrable or not at infinity, where F(s)=\int_0^s f(\gs)d\gs. We also give
new results dealing with non uniqueness for the initial value problem with
unbounded initial data. If , we prove that, for a large class of
nonlinearities , any positive solution admits an initial trace in the class
of positive Borel measures. As a model case we consider the case f(u)=u^\ga
\ln^\gb(u+1), where \ga>0 and \gb\geq 0
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