$L_p$-theory for a Cahn-Hilliard-Gurtin system
Mathias Wilke
Evolution Equations & Control Theory 2012, 1(2): 393-429 doi: 10.3934/eect.2012.1.393
In this paper we study a generalized Cahn-Hilliard equation which was proposed by Gurtin [9]. We prove the existence and uniqueness of a local-in-time solution for a quasilinear version, that is, if the coefficients depend on the solution and its gradient. Moreover we show that local solutions to the corresponding semilinear problem exist globally as long as the physical potential satisfies certain growth conditions. Finally we study the long-time behaviour of the solutions and show that each solution converges to a equilibrium as time tends to infinity.
keywords: quasilinear elliptic-parabolic system Lojasiewicz-Simon inequality. convergence to steady states Cahn-Hilliard-Gurtin equation global existence optimal regularity
Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces
Stefan Meyer Mathias Wilke
Evolution Equations & Control Theory 2013, 2(2): 365-378 doi: 10.3934/eect.2013.2.365
We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal $L_p$-regularity for parabolic equations and the implicit function theorem.
keywords: quasilinear parabolic system Kuznetsov's equation global well-posedness optimal regularity exponential decay.
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
Irena Lasiecka Mathias Wilke
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5189-5202 doi: 10.3934/dcds.2013.33.5189
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.
keywords: existence and uniqueness of strong solutions maximal regularity Quasilinear thermoelastic plates exponential decay.

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