Summary
Consider the system
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).
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Partially supported by Grant PGV9101 and CICYT Grant PB90-0245.
Partially supported by CICYT Grant PB90–0235.
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Escobedo, M., Herrero, M.A. A semilinear parabolic system in a bounded domain. Annali di Matematica pura ed applicata 165, 315–336 (1993). https://doi.org/10.1007/BF01765854
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DOI: https://doi.org/10.1007/BF01765854