The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations

Abstract

We study nonnegative solutions of the initial value problem for a weakly coupled system

$$\begin{gathered} \left\{ \begin{gathered} u_t = L_1 u + t^{s1} \upsilon ^{p1} \hfill \\ \upsilon _t = L_2 \upsilon + t^{s2} u^{p2} \hfill \\ \end{gathered} \right.(x,t) \in \mathbb{R}^N \times (0,T) \hfill \\ u(x,0) = u_0 (x) \geqslant 0 \hfill \\ \upsilon (x,0) = \upsilon _0 (x) \geqslant 0x \in \mathbb{R}^N \hfill \\ u_0 ,\upsilon _0 \in L^\infty (\mathbb{R}^N ) \hfill \\ \end{gathered} $$

wheres ₁,s ₂ ≥ 1 withp ₁ p ₂ > 1 and theL ₁,L ₂ are uniformly elliptic second order differential operators with uniformly bounded coefficients. It is proved that if

$$\max \left( {\frac{{(s_2 + 1)p_1 + s_1 + 1}}{{p_1 p_2 - 1}},\frac{{(s_1 + 1)p_2 + s_2 + 1}}{{p_1 p_2 - 1}}} \right) \geqslant \frac{N}{2}$$

then every nontrivial nonnegative solution is not global in time; whereas if

$$\max \left( {\frac{{(s_2 + 1)p_1 + s_1 + 1}}{{p_1 p_2 - 1}},\frac{{(s_1 + 1)p_2 + s_2 + 1}}{{p_1 p_2 - 1}}} \right)< \frac{N}{2}$$

then there exists both positive global solutions and nonglobal solutions. We obtain some results for the system

$$\left\{ \begin{gathered} u_t = \Delta u + \left| x \right|^{\sigma 1} \upsilon ^{p1} \hfill \\ \upsilon _t = \Delta \upsilon + \left| x \right|^{\sigma 2} u^{p2} \hfill \\ \end{gathered} \right.(x,t) \in \mathbb{R}^N \times (0,{\rm T})$$

where σ₁, σ₂ ≥ 0 andp ₁,p ₂ ≥ 1 withp ₁ p ₂ > 1.