Nonexistence results for a time-fractional biharmonic diffusion equation

Abstract

Abstract We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ∂ t α u + ∂ t β u + u x x x x = h ( t , x ) | u | p $\partial _{t}^{\alpha }u+\partial _{t}^{\beta }u+u_{xxxx}=h(t,x)|u|^{p}$ in ( 0 , ∞ ) × ( 0 , 1 ) $(0,\infty )\times (0,1)$ subject to the initial conditions u ( 0 , x ) = u 0 ( x ) $u(0,x)=u_{0}(x)$ , u t ( 0 , x ) = u 1 ( x ) $u_{t}(0,x)=u_{1}(x)$ and the Navier boundary conditions u ( t , 1 ) = u x x ( t , 1 ) = 0 $u(t,1)=u_{xx}(t,1)=0$ , where α ∈ ( 0 , 1 ) $\alpha \in (0,1)$ , β ∈ ( 1 , 2 ) $\beta \in (1,2)$ , ∂ t α $\partial _{t}^{\alpha}$ (resp. ∂ t β $\partial _{t}^{\beta}$ ) is the fractional derivative of order α (resp. β) with respect to the time-variable in the Caputo sense, p > 1 $p>1$ and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator ∂ 4 ∂ x 4 $\frac{\partial ^{4}}{\partial x^{4}}$ , the domain, the initial conditions and the boundary condition, a general nonexistence result is established. Next, some special cases of weight functions h are discussed.

Keywords

• Time-fractional biharmonic diffusion equation
• Weak solution
• Nonexistence
• Caputo fractional derivative