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Communications on Pure & Applied Analysis

May 2021 , Volume 20 , Issue 5

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Homogenization of a modified bidomain model involving imperfect transmission
Micol Amar, Daniele Andreucci and Claudia Timofte
2021, 20(5): 1755-1782 doi: 10.3934/cpaa.2021040 +[Abstract](308) +[HTML](139) +[PDF](472.67KB)

We study, by means of the periodic unfolding technique, the homogenization of a modified bidomain model, which describes the propagation of the action potential in the cardiac electrophysiology. Such a model, allowing the presence of pathological zones in the heart, involves various geometries and non-standard transmission conditions on the interface between the healthy and the damaged part of the cardiac muscle.

Partial regularity for parabolic systems with VMO-coefficients
Leon Mons
2021, 20(5): 1783-1820 doi: 10.3934/cpaa.2021041 +[Abstract](229) +[HTML](103) +[PDF](516.46KB)

In this article we establish a partial Hölder continuity result for weak solutions of parabolic systems, where the nonlinear vector field \begin{document}$ A(\cdot) $\end{document} satisfies a standard \begin{document}$ p $\end{document}-growth condition and a non-degenerate ellipticity condition with respect to the gradient variable, while in the space-time variable \begin{document}$ z = (x,t) $\end{document} it verifies a VMO-type condition. Thus, no continuity in the space-time variable is assumed. The proof is based on the method of \begin{document}$ \mathcal{A} $\end{document}-caloric approximation, applied on suitably chosen intrinsic cylinders.

A note on the energy transfer in coupled differential systems
Monica Conti, Lorenzo Liverani and Vittorino Pata
2021, 20(5): 1821-1831 doi: 10.3934/cpaa.2021042 +[Abstract](316) +[HTML](115) +[PDF](718.4KB)

We study the energy transfer in the linear system

made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter \begin{document}$ b $\end{document}.

Spatial asymptotics of mild solutions to the time-dependent Oseen system
Paul Deuring
2021, 20(5): 1833-1849 doi: 10.3934/cpaa.2021044 +[Abstract](250) +[HTML](136) +[PDF](383.9KB)

We consider mild solutions to the 3D time-dependent Oseen system with homogeneous Dirichlet boundary conditions, under weak assumptions on the data. Such solutions are defined via the semigroup generated by the Oseen operator in \begin{document}$ L^q. $\end{document} They turn out to be also \begin{document}$ L^q $\end{document}-weak solutions to the Oseen system. On the basis of known results about spatial asymptotics of the latter type of solutions, we then derive pointwise estimates of the spatial decay of mild solutions. The rate of decay depends in particular on \begin{document}$ L^p $\end{document}-integrability in time of the external force.

Jump and variational inequalities for averaging operators with variable kernels
Zhenbing Gong, Yanping Chen and Wenyu Tao
2021, 20(5): 1851-1866 doi: 10.3934/cpaa.2021045 +[Abstract](304) +[HTML](121) +[PDF](369.77KB)

In this paper, we prove that the jump function and variation of averaging operators with rough variable kernels are bounded on \begin{document}$ L^{2}(\mathbb{R}^{n}) $\end{document} if \begin{document}$ \Omega\in L^{\infty}(\mathbb{R}^{n})\times L^{q}(\mathbb{S}^{n-1}) $\end{document} for \begin{document}$ q>2(n-1)/n $\end{document} and \begin{document}$ n\geq2 $\end{document}. Moreover, we obtain the boundedness on weighted \begin{document}$ L^{p}(\mathbb{R}^{n}) $\end{document} spaces of the jump function and \begin{document}$ \rho $\end{document}-variations for averaging operators with smooth variable kernels. Finally, we extend the result to the Morrey spaces.

Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces
Tôn Việt Tạ
2021, 20(5): 1867-1891 doi: 10.3934/cpaa.2021050 +[Abstract](286) +[HTML](113) +[PDF](369.82KB)

This paper is devoted to studying stochastic semilinear evolution equations in Banach spaces of M-type 2. First, we prove existence, uniqueness and regularity of strict solutions. Then, we give an application to stochastic partial differential equations.

Forward-backward approximation of nonlinear semigroups in finite and infinite horizon
Andrés Contreras and Juan Peypouquet
2021, 20(5): 1893-1906 doi: 10.3934/cpaa.2021051 +[Abstract](326) +[HTML](134) +[PDF](397.04KB)

This work is concerned with evolution equations and their forward-backward discretizations, and aims at building bridges between differential equations and variational analysis. Our first contribution is an estimation for the distance between iterates of sequences generated by forward-backward schemes, useful in the convergence and robustness analysis of iterative algorithms of widespread use in numerical optimization and variational inequalities. Our second contribution is the approximation, on a bounded time frame, of the solutions of evolution equations governed by accretive (monotone) operators with an additive structure, by trajectories constructed by interpolating forward-backward sequences. This provides a short, simple and self-contained proof of existence and regularity for such solutions; unifies and extends a number of classical results; and offers a guide for the development of numerical methods. Finally, our third contribution is a mathematical methodology that allows us to deduce the behavior, as the number of iterations tends to \begin{document}$ +\infty $\end{document}, of sequences generated by forward-backward algorithms, based solely on the knowledge of the behavior, as time goes to \begin{document}$ +\infty $\end{document}, of the solutions of differential inclusions, and viceversa.

Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay
Wenjing Liu, Rong Yang and Xin-Guang Yang
2021, 20(5): 1907-1930 doi: 10.3934/cpaa.2021052 +[Abstract](284) +[HTML](137) +[PDF](409.69KB)

This paper is concerned with the pullback dynamics and asymptotic stability for a 3D Brinkman-Forchheimer equation with infinite delay. The well-posedness of weak solution to the 3D Brinkman-Forchheimer flow with infinite delay is investigated in the weighted space \begin{document}$ C_\kappa(H) $\end{document} firstly, then the pullback attractors are presented for the process of weak solution. Moreover, the existence of global attractor and the exponential stability analysis of stationary solutions are shown, which is based on the estimate of corresponding steady state equation.

Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body
Miroslav Bulíček, Victoria Patel, Yasemin Şengül and Endre Süli
2021, 20(5): 1931-1960 doi: 10.3934/cpaa.2021053 +[Abstract](361) +[HTML](136) +[PDF](468.2KB)

We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form \begin{document}$ \boldsymbol{u}_{tt} = \mbox{div }\mathbb{T} + \boldsymbol{f} $\end{document} for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor \begin{document}$ \boldsymbol{\varepsilon}( \boldsymbol{u}) $\end{document} to the Cauchy stress tensor \begin{document}$ \mathbb{T} $\end{document}, is assumed to be of the form \begin{document}$ \boldsymbol{\varepsilon}( \boldsymbol{u}_t) + \alpha \boldsymbol{\varepsilon}( \boldsymbol{u}) = F( \mathbb{T}) $\end{document}, where we define \(F(\mathbb{T}) = (1 + | \mathbb{T}|^a)^{-\frac{1}{a}} \mathbb{T}\), for constant parameters \begin{document}$ \alpha \in (0,\infty) $\end{document} and \begin{document}$ a \in (0,\infty) $\end{document}, in any number \begin{document}$ d $\end{document} of space dimensions, with periodic boundary conditions. The Cauchy stress \begin{document}$ \mathbb{T} $\end{document} is shown to belong to \begin{document}$ L^{1}(Q)^{d \times d} $\end{document} over the space-time domain \begin{document}$ Q $\end{document}. In particular, in three space dimensions, if \begin{document}$ a \in (0,\frac{2}{7}) $\end{document}, then in fact \begin{document}$ \mathbb{T} \in L^{1+\delta}(Q)^{d \times d} $\end{document} for a \begin{document}$ \delta > 0 $\end{document}, the value of which depends only on \begin{document}$ a $\end{document}.

Spectral properties of ordinary differential operators admitting special decompositions
Krzysztof Stempak
2021, 20(5): 1961-1986 doi: 10.3934/cpaa.2021054 +[Abstract](210) +[HTML](112) +[PDF](443.42KB)

We investigate spectral properties of ordinary differential operators related to expressions of the form \begin{document}$ D^{\epsilon}+a $\end{document}. Here \begin{document}$ a\in \mathbb{R} $\end{document} and \begin{document}$ D^{\epsilon} $\end{document} denotes a composition of \begin{document}$ \mathfrak{d} $\end{document} and \begin{document}$ \mathfrak{d}^+ $\end{document} according to the signs in the multi-index \begin{document}$ {\epsilon} $\end{document}, where \begin{document}$ \mathfrak{d} $\end{document} is a first order linear differential expression, called delta-derivative, and \begin{document}$ \mathfrak{d}^+ $\end{document} is its formal adjoint in an appropriate \begin{document}$ L^2 $\end{document} space. In particular, Sturm-Liouville operators that admit the decomposition of the type \begin{document}$ \mathfrak{d}^+\mathfrak{d}+a $\end{document} are considered. We propose an approach, based on weak delta-derivatives and delta-Sobolev spaces, which is particularly useful in the study of the operators \begin{document}$ D^{\epsilon}+a $\end{document}. Finally we examine a number of examples of operators, which are of the relevant form, naturally arising in analysis of classical orthogonal expansions.

Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation
Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti and Vinicius A. Peralta
2021, 20(5): 1987-2020 doi: 10.3934/cpaa.2021055 +[Abstract](253) +[HTML](111) +[PDF](490.61KB)

We are concerned with the transmission problem of nonlinear viscoelastic waves in a heterogeneous medium, establishing the well-posedness of solutions and the exponential stability of the related energy functional. We introduce an auxiliary problem to prove the exponential stability and the proof combines an observability inequality and microlocal analysis tools.

A generalized complex Ginzburg-Landau equation: Global existence and stability results
Simão Correia and Mário Figueira
2021, 20(5): 2021-2038 doi: 10.3934/cpaa.2021056 +[Abstract](235) +[HTML](93) +[PDF](337.3KB)

We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.

Global well-posedness for effectively damped wave models with nonlinear memory
Tayeb Hadj Kaddour and Michael Reissig
2021, 20(5): 2039-2064 doi: 10.3934/cpaa.2021057 +[Abstract](207) +[HTML](112) +[PDF](393.65KB)

In this paper, we study the Cauchy problem for a special family of effectively damped wave models with nonlinear memory on the right-hand side. Our goal is to prove global (in time) well-posedness results for Sobolev solutions. Due to the effective dissipation the model is parabolic like from the point of view of energy decay estimates of the corresponding linear Cauchy problem with vanishing right-hand side. For this reason there appears a Fujita type exponent as a threshold. Applying modern tools from Harmonic Analysis we prove several results by taking into consideration different regularity properties of the data.

Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition
Claudianor O. Alves and César T. Ledesma
2021, 20(5): 2065-2100 doi: 10.3934/cpaa.2021058 +[Abstract](224) +[HTML](110) +[PDF](468.8KB)

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem

where \begin{document}$ a, b\in \mathbb{R} $\end{document} with \begin{document}$ a<b $\end{document}, \begin{document}$ (-\Delta)^{\frac{1}{2}} $\end{document} denotes the fractional Laplacian operator and \begin{document}$ \mathcal{N}_{1/2} $\end{document} is the nonlocal operator that describes the Neumann boundary condition, which is given by

A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties
Antonio De Rosa and Domenico Angelo La Manna
2021, 20(5): 2101-2116 doi: 10.3934/cpaa.2021059 +[Abstract](180) +[HTML](145) +[PDF](372.05KB)

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the non local Gaussian perimeter taken into consideration.

Periodic solutions of p-Laplacian equations via rotation numbers
Shuang Wang and Dingbian Qian
2021, 20(5): 2117-2138 doi: 10.3934/cpaa.2021060 +[Abstract](226) +[HTML](100) +[PDF](457.47KB)

We investigate the existence and multiplicity of periodic solutions of the \begin{document}$ p $\end{document}-Laplacian equation \begin{document}$ \left(\phi_p(x')\right)'+f(t, x) = 0 $\end{document}. Both asymptotically linear and partially superlinear nonlinearities are studied, in absence of global existence and uniqueness conditions on the solutions of the associated Cauchy problems and the sign assumption on \begin{document}$ f $\end{document}. We use a approach of rotation number in the \begin{document}$ p $\end{document}-polar coordinates transformation, together with the phase-plane analysis of the rotational properties of large solutions and a recent version of Poincaré-Birkhoff theorem for Hamiltonian systems, for obtaining multiplicity results of \begin{document}$ p $\end{document}-Laplacian equation in terms of the gap between the rotation numbers of referred piecewise \begin{document}$ p $\end{document}-linear systems at zero and infinity.

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