Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.     

The Dirichlet Problem for the Total Variation Flow

We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.     

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

Abstract

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.     

A nonlocal diffusion problem on manifolds

ABSTRACT

In this paper we study a nonlocal diffusion problem on a manifold. These kinds of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.     

Weak solutions for a system of nonlinear Klein-Gordon equations

Summary We prove the existence and uniqueness of weak solutions of the mixed problem for a class of systems of nonlinear Klein-Gordon equations. Uniqueness is proved when the spatial dimension is either n=1, 2or 3.Partially supported by CNPq-Brasil.     

ON THE TWO-DIMENSIONAL VLASOV-HELMHOLTZ EQUATION WITH INFINITE MASS

ABSTRACT

We prove existence and uniqueness of the solutions to the Vlasov-Helmholtz equation in two dimensions in the case of unbounded mass.     

Existence and Uniqueness of Entropy Solution of Scalar Conservation Laws with a Flux Function Involving Discontinuous Coefficients

ABSTRACT

In this paper, the question of existence and uniqueness for entropy solutions of scalar conservation laws with a flux function which is discontinuous with respect to the space variable is investigated. We show that no extra assumption of convexity or genuine non-linearity with respect to the state variable of the flux function is required for the problem to be well-posed and prove it. The proof uses a kinetic formulation of the conservation law.     

On well-posedness of the semilinear heat equation on the sphere

We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L ^p ?? L ^q estimates for the semigroup generated by the Laplace?CBeltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n??? 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.     

Optimal control problem for cancer invasion parabolic system with nonlinear diffusion

ABSTRACT

In this paper, we study a distributed optimal control problem of a coupled nonlinear system of reaction–diffusion equations. The system consists of three partial differential equations to represent cancer cell density, matrix-degrading enzymes concentration and oxygen concentration, and an ordinary differential equation to describe the extracellular matrix concentration. Our aim is to minimize the growth of cancer cells by controlling the production of matrix-degrading enzymes. First, we prove the existence and uniqueness of solutions of the direct problem. Then, we prove the existence of an optimal control. Finally, we derive the first-order optimality conditions and prove the existence of weak solutions of the adjoint problem.     

On the Solvability of Some Dynamic Poroelastic Problems

We consider the direct problems for poroelasticity equations. In the low-frequency approximation we prove existence and uniqueness theorems for the solution to a certain mixed problem. In the high-frequency approximation we establish the uniqueness of a weak solution to the mixed problem and its continuous dependence on the data in the cases of bounded and unbounded temporal intervals and for however many spatial variables.

     

On Classical Solutions of the Nordström-Vlasov System

Abstract

The Nordström-Vlasov system describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordström scalar theory of gravitation. We prove existence and uniqueness of classical solutions of the Cauchy problem in three dimensions and establish a condition which guarantees that the solution is global in time. Moreover, we show that if one changes the sign of the source term in the field equation, which heuristically corresponds to the case of a repulsive gravitational force, then solutions blow up in finite time for a large class of initial data. Finally, we prove global existence of classical solutions for the one dimensional version of the system with the correct sign in the field equation.     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

On a generalized proximal point method for solving equilibrium problems in Banach spaces

We introduce a regularized equilibrium problem in Banach spaces, involving generalized Bregman functions. For this regularized problem, we establish the existence and uniqueness of solutions. These regularizations yield a proximal-like method for solving equilibrium problems in Banach spaces. We prove that the proximal sequence is an asymptotically solving sequence when the dual space is uniformly convex. Moreover, we prove that all weak accumulation points are solutions if the equilibrium function is lower semicontinuous in its first variable. We prove, under additional assumptions, that the proximal sequence converges weakly to a solution.     

Global Helically Symmetric Solutions to 3D MHD Equations

We prove the existence and uniqueness of global strong solutions to the Cauchy problem of the three-dimensional magnetohydrodynamic equations in R3 when initial data are helically symmetric. Moreover, the large-time behavior of the strong solutions is obtained simultaneously.     

A free boundary problem for a fluid flow in a heterogeneous porous medium

In this paper we study the problem of seepage of a fluid through a porous medium, assuming the flow governed by a nonlinear Darcy law and nonlinear leaky boundary conditions. We prove the continuity of the free boundary and the existence and uniqueness of minimal and maximal solutions. We also prove the uniqueness of theS ₃-connected solution in various situations.     

The well-posedness of bipolar semiconductor hydrodynamic model with recombination-generation rate on the bounded interval

ABSTRACT

We study the initial boundary value problem of bipolar semiconductor hydrodynamic model with recombination-generation rate for the non-constant doping profile. The new feature is that the current distribution for electrons and holes is not constant. In order to overcome this difficulty, the existence and uniqueness of a subsonic stationary solution are first established by elliptic theorem. Then, for such an Euler–Poisson system, we prove, by means of a technical energy method, that the subsonic solutions are unique, exist globally and asymptotically converge to the corresponding stationary solutions. An exponential decay rate is also derived.     

Existence of radial solutions for an asymptotically linear p-Laplacian problem

We prove that an asymptotically linear Dirichlet problem which involves the p-Laplacian operator has multiple radial solutions when the nonlinearity has a positive zero and the range of the ‘p-derivative’ of the nonlinearity includes at least the first j radial eigenvalues of the p-Laplacian operator. The main tools that we use are a uniqueness result for the p-Laplacian operator and bifurcation theory.     

Remark on compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial data

In this paper, we study the free boundary problem for 1D compressible Navier-Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier-Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as t→+∞.     

ON EXISTENCE AND UNIQUENESS IN A FREE BOUNDARY PROBLEM FROM COMBUSTION

ABSTRACT

We study a free boundary problem for the heat equation describing the propagation of laminar flames under certain geometric assumptions on the initial data. The problem arises as the limit of a singular perturbation problem, and generally no uniqueness of limit solutions can be expected. However, if the initial data is starshaped, we show that the limit solution is unique and coincides with the minimal classical supersolution. Under certain convexity assumption on the data, we prove first that the limit solution is a classical solution of the free boundary problem for a short time interval, and then that the solution, in fact, stays classical as long as it does not vanish identically.