Asymptotic behavior of a parabolic-hyperbolic system.

*(English)*Zbl 1079.35022The authors consider the asymptotic behaviour of the solution to the nonlinear parabolic-hyperbolic system of the following type
\[
\partial_t(\vartheta+\lambda(\chi))-\Delta\vartheta=0, \quad \mu\partial_{tt}\chi+ \partial_t\chi-\Delta\chi+ \chi+\phi(\chi)-\lambda'(\chi) \vartheta=g, \tag \(P_\mu\)
\]
on \(\Omega\times(0,\infty)\), with the boundary conditions \(\vartheta=\partial_n\chi=0\) on \(\partial\Omega\times(0,\infty)\) and the initial conditions \(\vartheta(0)=\vartheta_0\), \(\chi(0)=\chi_0\), \(\partial_t\chi(0)=\chi_1\) in \(\Omega,\partial_n\) being the outward normal derivative, where \(\Omega\subset\mathbb{R}^3\) is a bounded and connected domain with smooth boundary \(\partial\Omega\) occupied by a two-phase material for any time \(t\geq0\), \(\vartheta\) is the relative temperature, \(\chi\) is a nonconserved parameter with quadratic growth, \(\phi\) is a smooth function with cubic gorwth, \(\mu>0\) is a small inertial parameter. Assuming \(g\in L^2(\Omega)\) they proved that the problem \((P_\mu)\) generates a strongly continuous semigroup \(S_\mu(t)\) on the phase-space \(L^2(\Omega)\times H^1(\Omega)\times L^2(\Omega)\). Next they prove the existence of absorbing sets under some general assumptions on \(\Phi\). Using the energy estimate they deduced that the dissipation dynamical system \(S_\mu(t)\) has a universal attractor \(A_\mu\). Next, they prove the regularity property for \(A_\mu\) (they show that \(A_\mu\) is contained in a ball of \(H_0^1(\Omega)\times H^2(\omega)\times H^1(\Omega)\) with radius independent of \(\mu\)).

In order to do it, they use Gronwall’s lemmas, energy estimate and different approach in the proof of the existence of the universal attractor than the Kuratowski measure of noncompactness (which was given in the paper of the authors [Adv. Math. Sci. Appl. 13, 443–459 (2003; Zbl 1057.37068)]).

Basing on the obtained results they prove the existence of a universal attractor \(A_0\) bounded in \(H_0^1(\omega)\times H^2(\Omega)\) for the dynamical system \(S_0(t)\) on \(L^2(\Omega)\times H^1(\Omega)\) associated with the problem \[ \partial_t(\vartheta+\lambda(\chi))-\Delta\vartheta=0, \quad \partial_t\chi-\Delta\chi+ \chi+\phi(\chi)-\lambda'(\chi)\vartheta=0, \tag \(P_0\) \] in \(\Omega\times(0,\infty)\) with the boundary conditions \(\vartheta=\partial_n\chi=0\) on \(\partial\Omega\times(0,\infty)\) and the initial conditions \(\vartheta(0)=\vartheta_0\), \(\chi(0)=\chi_0\) in \(\Omega\). They prove that \(\lim_{\mu\to0}\text{dist}(A_\mu,A_0)=0,\) where “dist” denotes the Hausdorff semidistance.

At the end, they construct an exponential attractor. This construction gives them the possibility to prove that \(A_\mu\) has finite fractal dimension which is independent on \(\mu\) where \(\lambda\) is linear. This allows to deduce the existence of a exponential attractor also in the case \(\mu=0\). At last, they describe the application of the proved theorems to a semilinear wave equation of the form \[ \mu\partial_{tt}\chi+\partial_t\chi+A\chi+\phi(\chi)=g . \]

In order to do it, they use Gronwall’s lemmas, energy estimate and different approach in the proof of the existence of the universal attractor than the Kuratowski measure of noncompactness (which was given in the paper of the authors [Adv. Math. Sci. Appl. 13, 443–459 (2003; Zbl 1057.37068)]).

Basing on the obtained results they prove the existence of a universal attractor \(A_0\) bounded in \(H_0^1(\omega)\times H^2(\Omega)\) for the dynamical system \(S_0(t)\) on \(L^2(\Omega)\times H^1(\Omega)\) associated with the problem \[ \partial_t(\vartheta+\lambda(\chi))-\Delta\vartheta=0, \quad \partial_t\chi-\Delta\chi+ \chi+\phi(\chi)-\lambda'(\chi)\vartheta=0, \tag \(P_0\) \] in \(\Omega\times(0,\infty)\) with the boundary conditions \(\vartheta=\partial_n\chi=0\) on \(\partial\Omega\times(0,\infty)\) and the initial conditions \(\vartheta(0)=\vartheta_0\), \(\chi(0)=\chi_0\) in \(\Omega\). They prove that \(\lim_{\mu\to0}\text{dist}(A_\mu,A_0)=0,\) where “dist” denotes the Hausdorff semidistance.

At the end, they construct an exponential attractor. This construction gives them the possibility to prove that \(A_\mu\) has finite fractal dimension which is independent on \(\mu\) where \(\lambda\) is linear. This allows to deduce the existence of a exponential attractor also in the case \(\mu=0\). At last, they describe the application of the proved theorems to a semilinear wave equation of the form \[ \mu\partial_{tt}\chi+\partial_t\chi+A\chi+\phi(\chi)=g . \]

Reviewer: Jerzy Gawinecki (Warszawa)

##### MSC:

35B40 | Asymptotic behavior of solutions to PDEs |

35B45 | A priori estimates in context of PDEs |

35B41 | Attractors |