[Paper Review] Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains

Anton Savostianov|Durham Research Online (Durham University)|Mar 28, 2014

Advanced Mathematical Physics Problems16 citations

TL;DR

This paper establishes the existence of Shatah-Struwe solutions and a smooth exponential attractor for a sub-quintic wave equation with fractional damping $(- abla)^{eta} abla_t u$ in bounded domains, where $\alpha \in (0, \frac{1}{2})$. The key contribution is deriving Strichartz-type estimates for the linearized equation, enabling control of $L^5([0,T];L^{10}( abla))$ norms essential for proving well-posedness and attractor existence beyond the standard energy framework.

ABSTRACT

The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping damping term of the form $(-Δ)^α\partial_t u$, $α\in(0,\frac{1}{2})$, in bounded smooth domains of $\Bbb R^3$. It appears that to prove well-posedness and develop smooth attractor theory for the problem we need additional regularity of the solutions, which does not follow from the energy estimate. Considering the original problem as perturbation of the linear one the task is reduced to derivation of Strichartz type estimate for the linear wave equation with fractional damping, which is the main feature of the work. Existence of smooth exponential attractor for the natural dynamical system associated with the problem is also established.

Motivation & Objective

• To establish well-posedness of the semi-linear damped wave equation with fractional damping $(-\Delta)^\alpha \partial_t u$ for $\alpha \in (0, \frac{1}{2})$ in bounded smooth domains.
• To prove existence and uniqueness of Shatah-Struwe solutions, which require higher regularity than energy solutions.
• To develop smooth attractor theory for the dynamical system generated by the equation, extending results beyond the $\alpha \geq \frac{1}{2}$ regime.
• To overcome the lack of regularity in energy estimates by deriving Strichartz-type estimates for the linearized equation with fractional damping.
• To construct a smooth exponential attractor in the energy space $\mathcal{E}$, bounded in the higher regularity space $\mathcal{E}_1$.

Proposed method

• Treat the nonlinear problem as a perturbation of the linear damped wave equation with fractional damping $(-\Delta)^\alpha \partial_t u$.
• Derive Strichartz-type estimates for the linear wave equation with fractional damping in bounded domains, which control $L^5([0,T];L^{10}( abla))$ norms.
• Use the derived Strichartz estimates to prove existence and uniqueness of Shatah-Struwe solutions for sub-quintic nonlinearities.
• Establish boundedness and compactness of the global attractor in the energy space $\mathcal{E}$ using the absorbing property and smoothing effect.
• Apply the discrete dynamical system approach via the time-one map $S_1$ to construct an exponential attractor $\mathcal{M}$ in $\mathcal{E}$, leveraging the Lipschitz continuity of $S_1$ on a bounded absorbing set.
• Prove the key estimate $\|S_1\xi_1 - S_1\xi_2\|_{\mathcal{E}_\alpha} \leq L\|\xi_1 - \xi_2\|_{\mathcal{E}}$ for $\xi_1, \xi_2$ in the absorbing set, ensuring the existence of an exponential attractor.

Experimental results

Research questions

• RQ1 Can Strichartz-type estimates be established for the linear wave equation with fractional damping $(-\Delta)^\alpha \partial_t u$ in bounded smooth domains for $\alpha \in (0, \frac{1}{2})$?
• RQ2 Does the lack of regularity in standard energy estimates for $\alpha < \frac{1}{2}$ prevent the existence of Shatah-Struwe solutions for sub-quintic nonlinearities?
• RQ3 Can a smooth exponential attractor be constructed for the dynamical system generated by the semi-linear wave equation with fractional damping when $\alpha \in (0, \frac{1}{2})$?
• RQ4 Is the $L^5([0,T];L^{10}( abla))$ norm of solutions controllable via Strichartz estimates in this regime, enabling higher regularity beyond energy solutions?
• RQ5 Does the backward smoothing property on the weak attractor, known in the $\alpha = 0$ and $\alpha = \frac{1}{2}$ cases, extend to $\alpha \in (0, \frac{1}{2})$?

Key findings

• Strichartz-type estimates are derived for the linear wave equation with fractional damping $(-\Delta)^\alpha \partial_t u$ in bounded smooth domains for $\alpha \in (0, \frac{1}{2})$, enabling control of $L^5([0,T];L^{10}( abla))$ norms.
• Shatah-Struwe solutions exist and are unique for the semi-linear wave equation with sub-quintic nonlinearity $f(u)$ and $\alpha \in (0, \frac{1}{2})$, despite the absence of such regularity from energy estimates.
• A smooth exponential attractor $\mathcal{M}$ exists in the energy space $\mathcal{E}$, and it is bounded in the higher regularity space $\mathcal{E}_1 = H^{1+\alpha}_0(\Omega) \times H^\alpha(\Omega)$.
• The time-one map $S_1$ satisfies a Lipschitz estimate $\|S_1\xi_1 - S_1\xi_2\|_{\mathcal{E}_\alpha} \leq L\|\xi_1 - \xi_2\|_{\mathcal{E}}$ on a bounded absorbing set $\mathcal{B}$, which guarantees the existence of an exponential attractor.
• The global attractor $\mathcal{B} = \bigcup_{t \geq 0} S_t \mathcal{B}_R$ is compact in $\mathcal{E}$, positively invariant, and bounded in $\mathcal{E}_1$, ensuring the attractor's smoothness.
• The construction of the exponential attractor relies on the embedding $\mathcal{E}_\alpha \subset \mathcal{E}$ being compact and the uniform Lipschitz continuity of the semigroup $S_t$ on $\mathcal{B}$ in the $\mathcal{E}$-norm over $t \in [0,1]$.