JournalszaaVol. 25 , No. 2DOI 10.4171/zaa/1281

Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping

  • Nasser-edine Tatar

    King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
  • Mokhtar Kirane

    Université de la Rochelle, France
Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping cover

Abstract

We consider the nonlinear hyperbolic equation \begin{align*} u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x)\left| u\right| ^{p} \end{align*} posed in Q:=(0,)×RN,Q:=(0,\infty )\times \mathbb{R}^{N}, where D+αuD_{+}^{\alpha }u, % 0<\alpha <1 is a time fractional derivative, with given initial position and velocity u(0,x)=u0(x)u(0,x)=u_{0}(x) and ut(0,x)=u1(x).u_{t}(0,x)=u_{1}(x). We find the Fujita's exponent which separates in terms of p,αp,\alpha and N,N, the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on u1(x)u_{1}(x) and h(x,t)h(x,t) assuring non-existence of local solutions.