Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions

This paper concerns with blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space. We establish the rate estimates for blow-up solutions and prove that the blow-up set is under proper conditions on initial data. Furthermore, for N=1, more complete conclusions about such two topics are given.     

Optimal control of stochastic differential equations with dynamical boundary conditions

In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problems with nonstandard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with nonstandard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation.     

Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions

In this paper we prove the existence and uniqueness of a weak solution for a non-autonomous reaction-diffusion model with dynamical boundary conditions. After that, a continuous dependence result is established via an energy method, including in particular some compactness properties. Finally, the precedent results are used in order to ensure the existence of minimal pullback attractors in the frameworks of universes of fixed bounded sets and that given by a tempered growth condition. The relation among these families is also discussed.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions

Using the upper and lower solution techniques and Hopf's maximum principle, the sufficient conditions for the existence of blow-up positive solution and global positive solution are obtained for a class of quasilinear parabolic equations subject to Neumann boundary conditions. An upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified.     

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation − Δu = λ u − u ^p in Ω, with the nonlinear boundary condition ∂u/∂ν = u ^r on ∂Ω. Here Ω is a smooth bounded domain of with outward unit normal ν, λ is a real parameter and p, r > 0. We also give the precise behavior of solutions for large |λ| in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods.      

Heat equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the heat equation posed in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

Blow-up time and boundary layer for solutions in parabolic equations with different diffusion

This paper deals with parabolic equations with different diffusion coefficients and coupled nonlinear sources, subject to homogeneous Dirichlet boundary conditions. We give many results about blow-up solutions, including blow-up time estimates for all of the spatial dimensions, the critical non-simultaneous blow-up exponents, uniform blow-up profiles, blow-up sets, and boundary layer with or without standard conditions on nonlocal sources. The conditions are much weaker than the ones for the corresponding results in the previous papers.     

Hele-Shaw type problems with dynamical boundary conditions

In this paper, we study the nonlinear evolution equation of Hele-Shaw type with dynamical boundary conditions. That is, the equation ut=Δw+f where u∈H(w) and H is the Heaviside function, with boundary condition μ(x,w)t_∂w+k∇w⋅ν=g, where ν denotes the outward normal vector of the fixed boundary of the domain. We prove existence, uniqueness and some qualitative properties of the solution.     

Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches. This article was submitted by the authors in English.     

Almost periodic and almost automorphic solutions to semilinear parabolic boundary differential equations

In this paper, we study the existence and uniqueness of almost periodic and almost automorphic solutions to the semilinear parabolic boundary differential equations

where AA_m|kerL generates a hyperbolic analytic semigroup on a Banach space X. The functions h and are defined on some intermediate subspaces X_β,0<β<1, and take values in X and in a boundary space ∂X respectively.     

A linearized difference scheme for semilinear parabolic equations with nonlinear absorbing boundary conditions

A novel three level linearized difference scheme is proposed for the semilinear parabolic equation with nonlinear absorbing boundary conditions. The solution of this problem will blow up in finite time. Hence this difference scheme is coupled with an adaptive time step size, i.e., when the solution tends to infinity, the time step size will be smaller and smaller. Furthermore, the solvability, stability and convergence of the difference scheme are proved by the energy method. Numerical experiments are also given to demonstrate the theoretical second order convergence both in time and in space in L_∞-norm.     

Blow‐up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions

This paper deals with the blow‐up phenomena for a system of parabolic equations with nonlinear boundary conditions. We show that under some conditions on the nonlinearities, blow‐up occurs at some finite time. We also obtain upper and lower bounds for the blow‐up time when blow‐up occurs. Copyright © 2014 John Wiley & Sons, Ltd.     

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L²(Ω), and hence, the associated first order Cauchy problem is well posed on Lq(Ω) for every q∈[1,∞). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.     

On the Laplace equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the Laplace equation in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

Absence of local and global solutions to an elliptic system with time-fractional dynamical boundary conditions

This paper presents extensions of some nonexistence results for elliptic systems with dynamical boundary conditions involving the time-derivatives of integer orders to the case of noninteger order. In particular, we consider a system of Poisson’s equations with time-fractional derivatives of order less than one in the boundary conditions and specify the thresholds of the nonlinearities which lead to the absence of global solutions. The fractional derivatives here are meant in the Riemann-Liouville sense (or in the Caputo sense). We also present necessary conditions for the existence of local solutions.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

On a class of parabolic equations with nonlocal boundary conditions

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.