FRACTIONAL APPROXIMATIONS OF THERMOELASTIC PLATE SYSTEMS

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with $N\geqslant 2$, which boundary $\partial\Omega$ is assumed to be a $\mathcal{C}^4$-hypersurface. In this paper we consider the initial-boundary value problem associated with the following thermoelastic plate system \[ \begin{cases} \partial_t^2u +\Delta^2 u+\Delta\theta=f(u),\ & x\in\Omega,\ t>0, \\ \partial_t\theta-\Delta \theta-\Delta \partial_tu=0,\ & x\in\Omega,\ t>0, \end{cases} \] subject to boundary conditions \[ \begin{cases} u=\Delta u=0,\ & x\in\partial\Omega,\ t>0,\\ \theta=0,\ & x\in\partial\Omega,\ t>0, \end{cases} \] and initial conditions \[ u(x,0)=u_0(x),\ \partial_tu(x,0)=v_0(x)\ \mbox{and}\ \theta(x,0)=\theta_0(x),\ x\in\Omega. \] We calculate explicit the fractional powers of the thermoelastic plate operator associated with this system via Balakrishnan integral formula and we present a fractional approximated system. We obtain a result of local well-posedness of the thermoelastic plate system and of its fractional approximations via geometric theory of semilinear parabolic systems.

Asymptotic and quenching behaviors of semilinear parabolic systems with singular nonlinearities

<p style='text-indent:20px;'>In this paper, we consider a family of parabolic systems with singular nonlinearities. We study the classification of global existence and quenching of solutions according to parameters and initial data. Furthermore, the rate of the convergence of the global solutions to the minimal steady state is given. Due to the lack of variational characterization of the first eigenvalue to the linearized elliptic problem associated with our parabolic system, some new ideas and techniques are introduced.</p>

Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg–Landau equation

We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mané projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of three-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. The method combines the so-called spatial-averaging principle invented by Sell and Mallet–Paret with temporal averaging of rapid oscillations which come from cross-diffusion terms.

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

In this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].