A uniqueness result for a semilinear reaction-diffusion system

Abstract:

Let $(u(t,x),v (t,x))$ and $(\bar u(t,x),\bar v (t,x))$ be two nonnegative classical solutions of (S) \[ \left \{ {\begin {array}{*{20}{c}} {{u_t} = \Delta u + {v ^p},} & {p > 0} \\ {{v _t} = \Delta v + {u^q},} & {q > 0} \\ \end {array} } \right .\] in some strip ${S_T} = (0,T) \times {\mathbb {R}^N}$, where $0 < T \leq \infty$ , and suppose that \[ u(0,x) = \bar u(0,x),\quad v (0,x) = \bar v (0,x),\] 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 \[ u(t,x) = \bar u(t,x),\quad v (t,x) = \bar v (t,x)\quad {\text {in}}\;{S_T}.\] 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 \[ u(t;s) = {c_1}(t - s)_ + ^{(p + 1)/(1 - pq)},\quad v (t;s) = {c_2}(t - s)_ + ^{(q + 1)/(1 - pq)},\] where $0 \leq s \leq t,{c_1}$ and ${c_2}$ are two positive constants depending only on $p$ and $q$, and $(\xi )_+ = \max \{ \xi , 0 \}$.