$L^1-L^1$ Estimate for the Energy to Structurally Damped $\sigma$-Evolution Models with Time Dependent

Abstract

In this paper, we are interested in the $L^1-L^1$ estimate for the energy, the elastic energy $\Vert |D|^\sigma u(t,\cdot )\Vert _{L^1}$ and the kinetic energy $\Vert u_t(t,\cdot )\Vert _{L^1}$ to the Cauchy problems for a class of special time-dependent structurally damped $\sigma$-evolution models:

$$\begin{aligned} \left\{ \begin{array}{lll} u_{tt}+(-\Delta )^\sigma u+b(t)(-\Delta )^{\sigma /2} u_t=0,\quad (t,x)\in (0,\infty )\times {\mathbb {R}}^n,\\ u(0,x)=:u_0(x),\quad u_t(0,x)=:u_{1}(x),\quad \sigma >1,\quad x\in {\mathbb {R}}^n, \end{array} \right. \end{aligned}$$

(0.1)

where $b=b(t)$ is a positive decreasing function. We will study the decay rate of the energies for solution to the Cauchy problem for structural damped $\sigma$-evolution models with time-dependent structurally dissipations $b(t)(-\Delta )^\delta u_t$. These estimates rely on more structural properties of representations of solutions. We divide our considerations in to b(t) is strictly decreasing, that is, $b'(t)<0$ for $t>0$. By the explicit representation of the solution to the model (0.1) which allows to the radial symmetric, and to apply the theory of modified Bessel functions. Thanks to these two effects, we are able to obtain $L^1-L^1$ estimate for the energy of solution structurally $\sigma$-evolution problems. The main goal is to derive $L^p-L^q$ estimates not necessarily on the conjugate line for the elastic and kinetic energy of the solution to (0.1) in the following sense:

$$\begin{aligned} \Vert \partial _t^j|D|^{(1+(-1)^j)\sigma } u(t,\cdot )\Vert _{L^p}\lesssim C_0^j(t)\Vert u_0\Vert _{L^q}+C_1^j(t)\Vert u_1\Vert _{L^q}, \end{aligned}$$

for $1+\frac{1}{r}=\frac{1}{p}+\frac{1}{q},\quad j=0,1$. We are interested in explaining the behavior of the functions $C_0^j(t)$ and $C_1^j(t)$ for $j=0,1,\,\,t\rightarrow 0^+$ and $t\rightarrow \infty$.