Finite-time attractivity for semilinear tempered fractional wave equations

Abstract

We prove the existence and finite-time attractivity of solutions to semilinear tempered fractional wave equations with sectorial operator and superlinear nonlinearity. Our analysis is based on the α-resolvent theory, the fixed point theory for condensing maps and the local estimates of solutions. An application to a class of partial differential equations will be given.

Stability and asymptotics for fractional delay diffusion‐wave equations

“…By substituting the estimates ( 18), (19), and ( 20) into (17) and choosing a sufficiently large T 2 , we obtain…”

“…Generally speaking, the appearance of delay will change the original properties of the equations. Therefore, the qualitative properties, especially the stability research, are crucial for delay-dependent system (see [18][19][20]). Hence, in this paper, we consider the stability problem of the following time-fractional delay diffusion-wave equation C 𝜕 𝛼 t u(t, x) = Δu(t, x) + bu (t − 𝜏, x) , (x, t) ∈ Ω × (0, +∞),…”

“…Generally speaking, the appearance of delay will change the original properties of the equations. Therefore, the qualitative properties, especially the stability research, are crucial for delay‐dependent system (see [18–20]). Hence, in this paper, we consider the stability problem of the following time‐fractional delay diffusion‐wave equation $$$$ {}^C{\partial}_t^{\alpha }u\left(t,x\right)=\Delta u\left(t,x\right)+ bu\left(t-\tau, x\right),\left(x,t\right)\in \Omega \times \left(0,+\infty \right), $$$$ where $$$ 1<\alpha <2,t $$$ and $$$ x $$$ stand for time and space variables, respectively, $$$ \Omega $$$ is a bounded domain in $$$ {\mathrm{\mathbb{R}}}^N,\Delta $$$ presents the Laplace operator, $$$ b\in \mathrm{\mathbb{R}} $$$ is a constant, $$$ \tau \in {\mathrm{\mathbb{R}}}^{+} $$$ presents a time delay, and $$$ {}^C{\partial}_t^{\alpha } $$$ represents Caputo's fractional derivative operator.…”