Let us consider the following wave equation \begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \\ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\ y(0,x)=y_{0}(x)\text{ , }y_{t}(0,x)=y_{1}(x) & \text{in} & (0,1). \end{array} Assume for example that $\left( {{y_0},{y_1}} \right) \in H_0^1(0,1) \times {L^2}(0,1)$ (resp. $\left( {{y_0},{y_1}} \right) \in {L^2}(0,1) \times {H^{ - 1}}(0,1)$). In the case of time-dependent coefficients semigroups theory fails, we have to deal with this problem otherwise. I saw some books and I found that $a$ must be $C^1$ in time. Is this the optimal assumption? Because I could solve this problem by characteristics and I didn't need to this assumption, it was just $$a \in {L^2}((0,T) \times (0,1))$$. Any suggestions?. Thank you.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
Who said that semigroups theory fails? I think you can do this in several ways with the theory. For instance you can apply Theorem 5.3.2, pp. 168, Section 5.3 : Semilinear and Quasilinear evolution equations, in the book of Ahmed Nasir Uddin:
Semigroup theory with applications to systems and control, Longman Scientific & Technical, 1991.
-
1$\begingroup$ Thank you. I totally ignored that $\endgroup$– GustaveCommented Jan 12, 2019 at 10:25