Critical exponent for a damped wave system with fractional integral

Abstract

We shall present the critical exponent $$ F(p, q,\alpha):=\max\big\{\alpha+\frac{(\alpha+1)(p+1)}{pq-1}, \alpha+\frac{(\alpha+1)(q+1)}{pq-1}\big\}-\frac{1}{2} $$ for the Cauchy problem $$\displaylines{ u_{tt}-u_{xx}+u_t=J_{0|t}^{\alpha}(|v|^{p}), \quad (t,x)\in\mathbb{R}^{+}\times\mathbb{R},\cr v_{tt}-v_{xx}+v_t=J_{0|t}^{\alpha}(|u|^{q}), \quad (t, x)\in\mathbb{R}^{+}\times\mathbb{R},\cr (u(0,x), u_t(0,x))=(u_0(x),u_1(x)), \quad x\in \mathbb{R},\cr (v(0,x), v_t(0,x))=(v_0(x),v_1(x)), \quad x\in \mathbb{R},\cr }$$ where $p,q\geq 1$, $pq>1$ and $0<\alpha<1/2$; that is, the small data global existence of solutions can be derived to the problem above if $F(p, q, \alpha)<0$. Furthermore, in the case of $F(p, q, \alpha)\geq 0$ the non-existence of global solution can be obtained with the initial data having positive average value.

Keywords

• Damped wave equation
• fractional integral
• critical exponent
• global solution