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Discrete & Continuous Dynamical Systems - S

February 2021 , Volume 14 , Issue 2

Issue on recent progress in PDE: Theory and applications

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Thibaut Deheuvels, Antoine Henrot, El Haj Laamri, Alain Miranville, Jean Rodolphe Roche and Didier Schmitt
2021, 14(2): i-vi doi: 10.3934/dcdss.2020437 +[Abstract](215) +[HTML](106) +[PDF](467.0KB)
A brief tribute to pierre baras
Thibaut Deheuvels, Antoine Henrot, El Haj Laamri, Alain Miranville, Jean Rodolphe Roche and Didier Schmitt
2021, 14(2): vii-viii doi: 10.3934/dcdss.2020438 +[Abstract](232) +[HTML](115) +[PDF](259.75KB)
Stationary reaction-diffusion systems in $ L^1 $ revisited
El Haj Laamri and Michel Pierre
2021, 14(2): 455-464 doi: 10.3934/dcdss.2020355 +[Abstract](545) +[HTML](304) +[PDF](316.53KB)

We prove existence of \begin{document}$ L^1 $\end{document}-weak solutions to the reaction-diffusion system obtained as a stationary version of the system arising for the evolution of concentrations in a reversible chemical reaction, coupled with space diffusion. This extends a previous result by the same authors where restrictive assumptions on the number of chemical species are removed.

A generalization of a criterion for the existence of solutions to semilinear elliptic equations
Pierre Baras
2021, 14(2): 465-504 doi: 10.3934/dcdss.2020439 +[Abstract](254) +[HTML](117) +[PDF](579.6KB)

We prove an abstract result of existence of "good" generalized subsolutions for convex operators. Its application to semilinear elliptic equations leads to an extension of results by P.B-M.Pierre concerning a criterion for the existence of solutions to a semilinear elliptic or parabolic equation with a convex nonlinearity. We apply this result to the model problem \begin{document}$ -\Delta u = a |\nabla u|^p+ b|u|^q+f $\end{document} with Dirichlet boundary conditions where \begin{document}$ a,b>0 $\end{document}, \begin{document}$ p,q>1 $\end{document}. No other condition is made on \begin{document}$ p $\end{document} and \begin{document}$ q $\end{document}.

On classes of well-posedness for quasilinear diffusion equations in the whole space
Boris Andreianov and Mohamed Maliki
2021, 14(2): 505-531 doi: 10.3934/dcdss.2020361 +[Abstract](572) +[HTML](273) +[PDF](459.83KB)

Well-posedness classes for degenerate elliptic problems in \begin{document}$ {\mathbb R}^N $\end{document} under the form \begin{document}$ u = \Delta {{\varphi}}(x,u)+f(x) $\end{document}, with locally (in \begin{document}$ u $\end{document}) uniformly continuous nonlinearities, are explored. While we are particularly interested in the \begin{document}$ L^\infty $\end{document} setting, we also investigate about solutions in \begin{document}$ L^1_{loc} $\end{document} and in weighted \begin{document}$ L^1 $\end{document} spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of \begin{document}$ u\mapsto {{\varphi}}(x,u) $\end{document}. Under additional restrictions on the dependency of \begin{document}$ {{\varphi}} $\end{document} on \begin{document}$ x $\end{document}, we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity \begin{document}$ {{\varphi}}: {\mathbb R}\mapsto {\mathbb R} $\end{document}, the space \begin{document}$ L^\infty $\end{document} (endowed with the \begin{document}$ L^1_{loc} $\end{document} topology) is a well-posedness class for the problem \begin{document}$ u = \Delta {{\varphi}}(u)+f(x) $\end{document}.

The fast-sorption and fast-surface-reaction limit of a heterogeneous catalysis model
Björn Augner and Dieter Bothe
2021, 14(2): 533-574 doi: 10.3934/dcdss.2020406 +[Abstract](493) +[HTML](245) +[PDF](554.93KB)

Within this paper, we consider a heterogeneous catalysis system consisting of a bulk phase \begin{document}$ \Omega $\end{document} (chemical reactor) and an active surface \begin{document}$ \Sigma = \partial \Omega $\end{document} (catalytic surface), between which chemical substances are exchanged via adsorption (transport of mass from the bulk boundary layer adjacent to the surface, leading to surface-accumulation by a transformation into an adsorbed form) and desorption (the reverse process). Quite typically, as is the purpose of catalysis, chemical reactions on the surface occur several orders of magnitude faster than, say, chemical reactions within the bulk phase, and sorption processes are often quite fast as well. Starting from the non-dimensional version, different limit models, especially for fast surface chemistry and fast sorption at the surface, are considered. For a particular model problem, questions of local-in-time existence of strong and classical solutions and positivity of solutions are addressed.

Instability of free interfaces in premixed flame propagation
Claude-Michel Brauner and Luca Lorenzi
2021, 14(2): 575-596 doi: 10.3934/dcdss.2020363 +[Abstract](539) +[HTML](297) +[PDF](418.87KB)

In this survey, we are interested in the instability of flame fronts regarded as free interfaces. We successively consider a classical Arrhenius kinetics (thin flame) and a stepwise ignition-temperature kinetics (thick flame) with two free interfaces. A general method initially developed for thin flame problems subject to interface jump conditions is proving to be an effective strategy for smoother thick flame systems. It relies on the elimination of the free interface(s) and reduction to a fully nonlinear parabolic problem. The theory of analytic semigroups is a key tool to study the linearized operators.

Equipartition of energy for nonautonomous damped wave equations
Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein and Silvia Romanelli
2021, 14(2): 597-613 doi: 10.3934/dcdss.2020364 +[Abstract](532) +[HTML](259) +[PDF](353.92KB)

The kinetic and potential energies for the damped wave equation

\begin{document} $ \begin{equation} u''+2Bu'+A^2u = 0 \;\;\;\;\;\;({\rm DWE})\end{equation} $ \end{document}

are defined by

\begin{document}$ K(t) = \Vert u'(t)\Vert^2,\, P(t) = \Vert Au(t)\Vert^2, $\end{document}

where \begin{document}$ A,B $\end{document} are suitable commuting selfadjoint operators. Asymptotic equipartition of energy means

\begin{document}$\begin{equation} \lim\limits_{t\to\infty} \frac{K(t)}{P(t)} = 1 \;\;\;\;\;\;({\rm AEE})\end{equation}$ \end{document}

for all (finite energy) non-zero solutions of (DWE). The main result of this paper is the proof of a result analogous to (AEE) for a nonautonomous version of (DWE).

A mathematical model for marine dinoflagellates blooms
M. Dambrine, B. Puig and G. Vallet
2021, 14(2): 615-633 doi: 10.3934/dcdss.2020424 +[Abstract](289) +[HTML](147) +[PDF](412.23KB)

We present a model for the life cycle of a dinoflagellate in order to describe blooms. We prove the mathematical well-posedness of the model and the possibility of extinction in finite time of the alga form meaning that the full population is under the cysts from.

Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions
Klemens Fellner, Jeff Morgan and Bao Quoc Tang
2021, 14(2): 635-651 doi: 10.3934/dcdss.2020334 +[Abstract](655) +[HTML](322) +[PDF](361.61KB)

Uniform-in-time bounds of nonnegative classical solutions to reaction-diffusion systems in all space dimension are proved. The systems are assumed to dissipate the total mass and to have locally Lipschitz nonlinearities of at most (slightly super-) quadratic growth. This pushes forward the recent advances concerning global existence of reaction-diffusion systems dissipating mass in which a uniform-in-time bound has been known only in space dimension one or two. As an application, skew-symmetric Lotka-Volterra systems are shown to have unique classical solutions which are uniformly bounded in time in all dimensions with relatively compact trajectories in \begin{document}$ C(\overline{\Omega})^m $\end{document}.

A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions
Franck Davhys Reval Langa and Morgan Pierre
2021, 14(2): 653-676 doi: 10.3934/dcdss.2020353 +[Abstract](539) +[HTML](272) +[PDF](506.57KB)

We propose a time semi-discrete scheme for the Caginalp phase-field system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution is shown to converge to the energy solution of the problem as the time step tends to \begin{document}$ 0 $\end{document}. The proof involves a multivalued operator and a monotonicity argument. This approach allows us to compute numerically singular solutions to the problem.

Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties
Philippe Laurençot and Christoph Walker
2021, 14(2): 677-694 doi: 10.3934/dcdss.2020360 +[Abstract](533) +[HTML](269) +[PDF](338.8KB)

The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.

Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation
Patrick Martinez and Judith Vancostenoble
2021, 14(2): 695-721 doi: 10.3934/dcdss.2020362 +[Abstract](529) +[HTML](265) +[PDF](409.33KB)

We consider a reaction-diffusion model of biological invasion in which the evolution of the population is governed by several parameters among them the intrinsic growth rate \begin{document}$ \mu(x) $\end{document}. The knowledge of this growth rate is essential to predict the evolution of the population, but it is a priori unknown for exotic invasive species. We prove uniqueness and unconditional Lipschitz stability for the corresponding inverse problem, taking advantage of the positivity of the solution inside the spatial domain and studying its behaviour near the boundary with maximum principles. Our results complement previous works by Cristofol and Roques [11,13].

Theoretical and numerical analysis of a class of quasilinear elliptic equations
Nahed Naceur, Nour Eddine Alaa, Moez Khenissi and Jean R. Roche
2021, 14(2): 723-743 doi: 10.3934/dcdss.2020354 +[Abstract](677) +[HTML](277) +[PDF](944.24KB)

The purpose of this paper is to give a result of the existence of a non-negative weak solution of a quasilinear elliptic equation in the N-dimensional case, \begin{document}$ N\geq 1 $\end{document}, and to present a novel numerical method to compute it. In this work, we assume that the nonlinearity concerning the derivatives of the solution are sub-quadratics. The numerical algorithm designed to compute an approximation of the non-negative weak solution of the considered equation has coupled the Newton method with domain decomposition and Yosida approximation of the nonlinearity. The domain decomposition is adapted to the nonlinearity at each step of the Newton method. Numerical examples are presented and commented on.

A semilinear heat equation with initial data in negative Sobolev spaces
Haruki Umakoshi
2021, 14(2): 745-767 doi: 10.3934/dcdss.2020365 +[Abstract](651) +[HTML](298) +[PDF](464.94KB)

We give a sufficient conditions for the existence, locally in time, of solutions to semilinear heat equations with nonlinearities of type \begin{document}$ |u|^{p-1}u $\end{document}, when the initial datas are in negative Sobolev spaces \begin{document}$ H_q^{-s}(\Omega) $\end{document}, \begin{document}$ \Omega \subset \mathbb{R}^N $\end{document}, \begin{document}$ s \in [0,2] $\end{document}, \begin{document}$ q \in (1,\infty) $\end{document}. Existence is for instance proved for \begin{document}$ q>\frac{N}{2}\left(\frac{1}{p-1}-\frac{s}{2}\right)^{-1} $\end{document}. This is an extension to \begin{document}$ s \in (0,2] $\end{document} of previous results known for \begin{document}$ s = 0 $\end{document} with the critical value \begin{document}$ \frac{N(p-1)}{2} $\end{document}. We also observe the uniqueness of solutions in some appropriate class.

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