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PDEs on spheres have many important applications in physical phenomena, oceanography and meteorology, geophysics. In this paper, we study the parabolic systems with Caputo-Fabrizio derivative. In order to establish the existence of the mild solution, we use the Banach fixed point theorem and some analysis of Fourier series associated with several evaluations of the spherical harmonics function. Some of the techniques on upper and lower bounds of the Mittag-Lefler functions are also applied. This is one of the first research results on the systems of parabolic diffusion on the sphere.

The main concern of this article is to investigate the boundary controllability of some $$2\times 2$$ one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components.

International audience; The main goal of this paper is to investigate the boundary controllability of some coupled parabolic systems in the cascade form in the case where the boundary conditions are of Robin type. In particular, we prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, that let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalisation method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems.

In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown. This research was partially supported by the project MTM2014-52859-P and by the Aragón Government and European Social Fund, Spain (group E24–17R ).

We study the following Keller-Segel chemotaxis system with logistic source and nonlinear secretion: \begin{align*} u_t=\Delta u- \nabla\cdot(u\nabla v)+\kappa(|x|)u-\mu(|x|)u^p\quad\text{and}\quad 0=\Delta v-v+u^\gamma, \end{align*} where $\kappa(\cdot),~\mu(\cdot):[0,R]\rightarrow [0,\infty)$, $\gamma\in (1,\infty)$, $p\in(\gamma+1,\infty)$ and $\Omega \subset \mathbb{R}^n, n\geq 2$. For this system, we prove the global existence of solutions under suitable assumptions on the initial condition and the functions $\kappa(\cdot)$ and $\mu(\cdot).$

We establish a new regularity property for weak solutions of linear parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally Holder continuous L p valued functions for some p > 2 .

We study gradient flows of integral functionals in noncylindrical bounded domains E subset of R-n [0, T). The systems of differential equations take the form partial derivative(t)u - divD(xi)f (x, u, Du) = -D(u)f (x, u, Du) on E, for an integrand f(x, u, Du) that is convex and coercive with respect to the W-1,W-P-norm for p > 1. We prove the existence of variational solutions on noncylindrical domains under the only assumption that Ln+1(partial derivative E) = 0, even for functionals that do not admit a growth condition from above. For nondecreasing domains, the solutions are unique and admit a time-derivative in L-2(E). For domains that decrease the most with bounded speed and integrands that satisfy a p-growth condition, we prove that the constructed solutions are continuous in time with respect to the L-2-norm and solve the above system of differential equations in the weak sense. Under the additional assumption that the domain also increases the most at finite speed, we establish the uniqueness of solutions.

We study degenerate and singular parabolic equations in one space dimension. The emphasis is put on the regularity of solutions and the creation as well as the evolution of facets. Facets are understood as flat parts of the graph of solutions being a result of extremely high singularity. The systems, which we consider, arise from the theory of crystals.

Necessary and sufficient conditions for global existence of classical solutions to a class of cross diffusion systems on n-dimensional domains are given. Examples of blow up solutions are also presented.

This paper focuses on the boundary approximate controllability of two classes of linear parabolic systems, namely a system of $n$ heat equations coupled through constant terms and a $2 \times 2$ cascade system coupled by means of a first order partial differential operator with space-dependent coefficients. &nbsp For each system we prove a sufficient condition in any space dimension and we show that this condition turns out to be also necessary in one dimension with only one control. For the system of coupled heat equations we also study the problem on rectangle, and we give characterizations depending on the position of the control domain. Finally, we prove the distributed approximate controllability in any space dimension of a cascade system coupled by a constant first order term. &nbsp The method relies on a general characterization due to H.O. Fattorini.

We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{��}lder continuous Lp valued functions for some p > 2., 23 pages. Proof of Lemma 3.3 corrected. Final version to appear in J. Math. Pures Appl

We investigate a nonstandard phase field model of Cahn-Hilliard type. The model describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in the papers arXiv:1103.4585 and arXiv:1109.3303 for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type., Key words: nonlinear phase field systems, Cahn-Hilliard systems, parabolic systems, optimal boundary control, first-order necessary optimality conditions. The interested reader can also see the preprint arXiv:1106.3668 where a distributed optimal control problem is studied for a similar system. arXiv admin note: significant text overlap with arXiv:1106.3668

International audience; Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the L 2 norm of the solution. This method is appropriate for the equations with time dependent coefficients. It yields new results and is applicable when the usual linearization method is not applicable.

We establish a partial regularity result for weak solutions of nonsingular parabolic systems with subquadratic growth of the type $$ \partial_t u - \mathrm{div} a(x,t,u,Du) = B(x,t,u,Du), $$ where the structure function $a$ satisfies ellipticity and growth conditions with growth rate $\frac{2n}{n+2} < p < 2$. We prove Hölder continuity of the spatial gradient of solutions away from a negligible set. The proof is based on a variant of a harmonic type approximation lemma adapted to parabolic systems with subquadratic growth.

We consider a problem of partial reconstruction of the source term, together with the solution, in a linear parabolic Cauchy problem in R m + n . The supplementary information, which is necessary to solve it, is given by the knowledge of ∫ R n u ( t , x , y ) d μ ( y ) for every ( t , x ) , where u is the solution of the parabolic problem, and μ is a complex valued Borel measure in R n .

We analyze stability property of a class of linear parabolic systems via static feedback. Stabilization via static feedback scheme is most difficult and challenging when both actuators and observation weights admit spillovers. This arises typically in the boundary observation–boundary feedback scheme. We propose a simple static feedback law containing a parameter γ, and enhance the stability property or achieve (slightly) stabilization. In some situations, the evolution of the substructure of finite dimension contains singularities regarding γ. We show that these singularities are removed as long as the dimension is not large.

Nonlinear parabolic systems of partial differential equations are considered. In a recent work, we have proposed a new iterative method based on the eigenfunction expansion to integrate these systems. In this paper, we prove the convergence of the method on bounded time intervals under certain condition that can be more easily to satisfy. We then show that the solution obtained by the new method will converge to the exact solution for a problem in combustion theory. Moreover, we determine the number of iterations needed to obtain a solution with a predetermined level of accuracy. It is expected that the convergence analysis can be used for similar systems of time dependence.

The purpose of this survey paper is to give a brief review of certain aspects of stability of norms and subnorms acting on algebras over a field F , either R or C . A norm N on an associative algebra A over F shall be called stable if for some positive constant σ, N ( a m ) ⩽ σ N ( a ) m for all a ∈ A , m = 1 , 2 , 3 … A norm shall be called strongly stable if the above inequality holds with σ = 1. We begin the paper by discussing several results regarding norm stability, including conditions under which norms on certain algebras are stable. The second part of the paper is devoted to applications, where we employ the notion of norm stability to obtain criteria for the convergence of a well-known family of finite-difference schemes for the initial-value problem associated with the parabolic system ∂ u ( x , t ) ∂ t = ∑ 1 ⩽ j ⩽ k ⩽ s A jk ∂ 2 u ( x , t ) ∂ x j ∂ x k + ∑ 1 ⩽ j ⩽ s B j ∂ u ( x , t ) ∂ x j + Cu ( x , t ) , where Ajk, Bj and C are constant matrices, Ajk being Hermitian. The third and last part of the paper deals with the question of stability for subnorms acting on subsets of power-associative algebras that are closed under scalar multiplication and under raising to powers. A subnorm f on such a set S is a real-valued function satisfying f(a) > 0 for all 0 ≠ a ∈ S , and f(αa) = ∣α∣f(a) for all a ∈ S and α ∈ F .

In this paper we investigate the zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W_{t}(x,t) + (1/epsilon) A(x,D) W(x,t) = (1/epsilon^2) B(x,W(x,t)) + (1/epsilon) D(W(x,t)) + E(W(x,t)). We analyse the singular convergence, as epsilon tends to 0, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce ``energy estimates'', uniformly in epsilon, by assuming the existence of a symmetrizer having the so called block structure and by assuming ``dissipativity conditions'' on B. (iii) We perform the convergence analysis by using generalizations of Compensated Compactness due to Tartar and Gerard. Finally we include examples which show how to use our theory to approximate prescribed general quasilinear parabolic systems, satisfying Petrowski parabolicity condition, or general reaction diffusion systems., Comment: 26 pages, preliminary version Dec.00

The existence of a Nash equilibrium feedback is established for a two-player nonzero-sum stochastic differential game with discontinuous feedback. This is obtained by studying a parabolic system strongly coupled by discontinuous terms.

This paper deals with the blow-up rate estimates of positive solutions for semilinear parabolic systems with nonlinear boundary conditions. The upper and lower bounds of blow-up rates are obtained.

We study enhancement of stability or stabilization of a class of linear parabolic systems via static feedback. Static feedback scheme is most difficult when both actuators and observation weights admit spillovers. We propose a simple static feedback law of enhancing stability property or achieving stabilization.

Hysteresis operators have recently proved to be a powerful tool in modelling phase transition phenomena which are accompanied by the occurrence of hysteresis effects. In a series of papers, the present authors have proposed phase-field models in which hysteresis non-linearities occur at several places. A very important class of hysteresis operators studied in this connection is formed by the so-called Prandtl–Ishlinskii operators. For these operators, the corresponding phase-field systems are in the multi-dimensional case only known to admit unique solutions if the characteristic convex sets defining the operators are polyhedrons. In this paper, we use approximation techniques to extend the known results to multi-dimensional Prandtl–Ishlinskii operators having non-polyhedral convex characteristicsets. Copyright © 2002 John Wiley & Sons, Ltd.

The aim of this paper is the introduction of a new concept that concerned the analysis of a large class of distributed parabolic systems. It is the general concept of gradient remediability. More precisely, we study with respect to the gradient observation, the existence of an input operator (gradient efficient actuators) ensuring the compensation of known or unknown disturbances acting on the considered system. Then, we introduce and we characterize the notions of exact and weak gradient remediability and their relationship with the notions of exact and weak gradient controllability. Main properties concerning the notion of gradient efficient actuators are considered. The minimum energy problem is studies, and we show how to find the optimal control, which compensates the disturbance of the system. Approximations and numerical simulations are also presented.Keywords: actuators efficient; disturbance; gradient; parabolic systems; remediability; sensors

Some generalized maximum principles are established for linear second-order parabolic systems in which both first-order and zero-order terms are coupled.

Motivated by the classical model for the binary alloy solidification (crystallization) problem, we show the local in time existence and uniqueness of solutions to a parabolic system strongly coupled through free boundary conditions of Stefan type. Using a modi¯cation of the standard change of variables method and coercive estimates in a weighted HÄolder space (the weight being a power of t) we obtain solutions with maximal global regularity (having at least equal regularity for t > 0 as at the initial moment). FCT, POCTI/MAT/34471/2000

This paper is concerned with phase-field systems of Penrose-Fife type which model the dynamics of a phase transition with non-conserved vectorial order parameter. The main novelty of the model is that the evolution of the order parameter vector is governed by a system consisting of one partial differential equation and one partial differential inclusion, which in the simplest case may be viewed as a diffusive approximation of the so-called multi-dimensional stop operator, which is one of the fundamental hysteresis operators. Results concerning existence, uniqueness and continuous dependence on data are presented which can be viewed as generalizations of recent results by the authors to cases where a diffusive hysteresis occurs.

Starting from sufficient conditions for regularity of weak solutions to quasilinear parabolic systems, uecessary conditions for loss of regularity are formulated. It is shown numerically that in some situations loss of regularity ("blow up") really happens accordingly to these conditions.

Auscher proved Gaussian upper bound estimates for the fundamental solutions to parabolic equations with complex coefficients in the case when coefficients are time-independent and a small perturbation of real coefficients. We prove the equivalence between the local boundedness property of solutions to a parabolic system and a Gaussian upper bound for its fundamental matrix. As a consequence, we extend Auscher's result to the time dependent case.