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.


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 $\endgroup$ – Gustave Jan 12 at 10:25

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.