Throughout this paper, it is always assumed that
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It is shown in [14] that every bounded solution of (1.1)-(1.3) on
decays to zero as
and behaves like the heat kernel. We give the large time behavior for higher dimensional case in the following theorem. In what follows,
represents the usual
-norm.
For nonnegative solutions of (1.1)-(1.3), we need (1.6) only for , because integrating (1.4) and (1.5) on
respectively, we observe that
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Concerning the existence of bounded solutions, it is possible for a nonnegative solution of (1.1)-(1.3) in
to blow up in finite time(see [2]). For a simplified version of (1.1)-(1.2) replaced (1.2) by
in
, it is shown that the nonnegative solution exists globally in time under the condition
(see [3]), and that the finite-time blowup of nonnegative solutions may occur under the condition
(see [3,12]). It is also shown in [12] that the finite-time blowup of nonnegative solutions in
may occur even if
is small. For blowup problems to (1.1), (1.2) in a bounded domain, we refer to [1,4,5,7,8,9,13,15,16,18,19] and the references therein.
The following theorem gives the existence of bounded solutions to (1.1)-(1.3) in
, when
are small but
is not necessarily small. The uniqueness of solutions is also given in the theorem. The nonnegativity of solutions is not assumed.
(ii) Let . For any given
let
be an initial function satisfying
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