Nonautonomous wave equation of memory type

Question

I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation $$u'' - \Delta u + \int\limits_0^t {g(t-s)} \Delta u(s)ds = 0$$ This problem can be written under the standard form of Cauchy problem $$U' = A(t)U$$ where$$ A(t)=\left( \begin{array}{cc} 0 & 1 \\ \Delta -\int_{0}^{t}g(t-s)\Delta ds & 0% \end{array}% \right) $$ It is obvious that we can't apply the classical semigroups approach because the operator $A$ in this case depends on $t$. I tried to find some references which talk about these things but I didn't secceed. I want from you some advice or halp. Thank you.

Answer 1

I have seen this kind of problem with $\int\limits_0^t {g(t-s)} \Delta u(s)ds=(g*\Delta u)(t)$ instead of $\int\limits_0^t {g(s)} \Delta u(s)ds$.

For this kind of convolution term, we have this paper where the authors study the more general case $$u''(t) + A u(t) -(g*Au)(t) = 0$$ with $A$ being a self adjoint positive definite operator. The idea is to take the solution $r$ of $$r(t)-(g*r)(t)=g(t)$$ and introduce the new variable $$v(t)=u(t)-(g*u)(t).$$

From this you obtain an equivalent problem, which also have a convolution term. Removing this term, you obtain an equation which can be studied via classical semigroup theory. This enables you to study the complete equation, via fixed-point method.

Here there is another paper, where the same idea is applied to Timoshenko beams.

(The said papers have free versions available on the internet; just ask google.)