ISSN:
1618-1891
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary Consider the system (S) $$\left\{ \begin{gathered} u_t - \Delta u = v^p , in Q = \{ (t, x), t 〉 0, x \in \Omega \} , \hfill \\ v_t - \Delta v = u^q , in Q , \hfill \\ u(0, x) = u_0 (x) v(0, x) = v_0 (x) in \Omega , \hfill \\ u(t, x) = v(t, x) = 0 , when t \geqslant 0, x \in \partial \Omega , \hfill \\ \end{gathered} \right.$$ where Ω is a bounded open domain in ℝN with smooth boundary, p and q are positive parameters, and functions u0 (x), v0(x) are continuous, nonnegative and bounded. It is easy to show that (S) has a nonnegative classical solution defined in some cylinder QT=(0, T) × Ω with T ⩽ ∞. We prove here that solutions are actually unique if pq ⩾ 1, or if one of the initial functions u0, v0 is different from zero when 0 〈 pq 〈 1. In this last case, we characterize the whole set of solutions emanating from the initial value (u0, v0)=(0, 0). Every solution exists for all times if 0〈pq⩽1, but if pq 〉 1, solutions may be global or blow up in finite time, according to the size of the initial value (u0, v0).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01765854
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