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A uniqueness result for a semilinear reaction-diffusion system


Authors: M. Escobedo and M. A. Herrero
Journal: Proc. Amer. Math. Soc. 112 (1991), 175-185
MSC: Primary 35K57
DOI: https://doi.org/10.1090/S0002-9939-1991-1043410-9
MathSciNet review: 1043410
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Abstract: Let $ (u(t,x),v (t,x))$ and $ (\bar u(t,x),\bar v (t,x))$ be two nonnegative classical solutions of (S)

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {{u_t} = \Delta u + {v ^p},} & {p > 0} \\ {{v _t} = \Delta v + {u^q},} & {q > 0} \\ \end{array} } \right.$

in some strip $ {S_T} = (0,T) \times {\mathbb{R}^N}$, where $ 0 < T \leq \infty $ , and suppose that

$\displaystyle u(0,x) = \bar u(0,x),\quad v (0,x) = \bar v (0,x),$

where $ u(0,x)$ and $ v (0,x)$ are continuous, nonnegative, and bounded real functions, one of which is not identically zero. Then one has

$\displaystyle u(t,x) = \bar u(t,x),\quad v (t,x) = \bar v (t,x)\quad {\text{in}}\;{S_T}.$

If $ pq \geq 1$, the result is also true if $ u(0,x) = v (0,x) = 0$. On the other hand, when $ 0 < pq < 1$, the set of solutions of (S) with zero initial values is given by

$\displaystyle u(t;s) = {c_1}(t - s)_ + ^{(p + 1)/(1 - pq)},\quad v (t;s) = {c_2}(t - s)_ + ^{(q + 1)/(1 - pq)},$

where $ 0 \leq s \leq t,{c_1}$ and $ {c_2}$ are two positive constants depending only on $ p$ and $ q$, and $ (\xi)_+ = \max \{ \xi, 0 \}$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1043410-9
Keywords: Reaction diffusion systems, uniqueness
Article copyright: © Copyright 1991 American Mathematical Society