Existence of Global Solutions for Systems of Reaction-diffusion Equations on Unbounded Domains
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We consider, an initial-value problem for the thermal-diffusive combustion system
ut = a∆u - uh(v)
vt = b∆u + d∆v + uh(v),
where a > 0, d > 0, b ≠ 0, x ∈ ℝn, n ≥ 1, with h(v) = vm, m is an even nonnegative integer, and the initial data u0, v0 are bounded uniformly continuous and nonnegative. It is known that by a simple comparison if b = 0, α = 1, d ≤ 1, and h(v) = vm with m ∈ ℕ*, the solutions are uniformly bounded in time. When d > a = 1, b = 0, h(v) = vm with m ∈ ℕ*, Collet and Xin  proved the existence of global classical solutions and showed that the L∞ norm of v can not grow faster than 0(log log t) for any space dimension. In our case, no comparison principle seems to apply. Nevertheless using techniques form , we essentially prove the existence of global classical solutions if a < d, b < 0, and v0 ≥ b/ a-d u0.
CitationBadraoui, S. (2002). Existence of global solutions for systems of reaction-diffusion equations on unbounded domains. Electronic Journal of Differential Equations, 2002(74), pp. 1-10.
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