Wavelet–Galerkin solutions of one dimensional elliptic problems

Daubechies wavelet bases are used for numerical solution of partial differential equations of one dimension by Galerkin method. Galerkin bases are constructed from Daubechies functions which are compactly supported and which constitute an orthonormal basis of L²(R)$L^2(\mathbb{R})$. Theoretical and numerical results are obtained for elliptic problems of second order with different types of boundary conditions. Optimal error estimates are also obtained. Comparison of solutions with simple finite difference method suggests that for this class of problems, the present method will provide a better alternative to other classical methods. The methodology can be generalized to multidimensional problems by taking care of some technical facts.     

Pseudo almost periodic solutions to parabolic boundary value inverse problems

We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.     

When are solutions to nonlinear elliptic boundary value problems convex?

We extend and apply a concavity maximum principle from [10, 9, 7] to some nonlinear elliptic boundary problems and free boundary problems on convex domains Ω?IRⁿ. In particular we extend "convex dead core' results from n = 2 as in [4 ] to arbitrary n. We also show the convexity of the coincidence set in the obstacle problem under suitable assumptions.     

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

In this paper we consider the existence of positive solution for some semilinear elliptic equations with Neumann boundary condition involving a critical Hardy–Sobolev exponent and Hardy terms with boundary singularities. Using mountain pass lemma without (PS) condition and the strong maximum principle, we get the existence of a positive solution.     

On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.     

On a parabolic–elliptic chemotactic model with coupled boundary conditions

This paper deals with a nonlinear system of parabolic–elliptic type with a logistic source term and coupled boundary conditions related to pattern formation. We prove the existence of a unique positive global in time classical solution. We also analyze the associated stationary problem. Moreover it is proved, under the assumption of sufficiently strong logistic damping, that there is only one nonzero homogeneous equilibrium, and all the solutions to the nonstationary problem tend to this steady state for large times.     

Noncoercive convection–diffusion elliptic problems with Neumann boundary conditions

We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation

The attractor of a parabolic–elliptic system

Following Coclite, Holden and Karlsen [G.M. Coclite, H. Holden and K.H. Karlsen, Well-posedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst. 13 (3) (2005) 659–682] and Tian and Fan [Lixin Tian, Jinling Fan, The attractor on viscosity Degasperis-Procesi equation, Nonlinear Analysis: Real World Applications, 2007], we study the dynamical behaviors of the parabolic–elliptic system

_ut+_(f(t,x,u))x+g(t,x,u)+_Px−ε_uxx=0$u_t+{\left(f(t,x,u)\right)}_x+g(t,x,u)+P_x-\epsilon u_{xx}=0$

Sample solutions of stochastic boundary value problems ∗

We prove existence theorems for nonlinear stochastic Sturm-Liouville problems which improve results from [4]. In the simplest case this is done by means of a known result about measurable selections of multivalued maps and a new fixed point theorem for stochastic nonlinear operators which is more realistic than existing ones     

Convergence of approximate solutions to an elliptic–parabolic equation without the structure condition

We study the Cauchy–Dirichlet problem for the elliptic–parabolic equation $$b(u)_t + {\rm div} F(u) - \Delta u = f$$ in a bounded domain. We do not assume the structure condition $$b(z) = b(\hat z) \Rightarrow F(z) = F(\hat z).$$ Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold (Proc R Soc Edinb 133A(3):477–496, 2003) where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux F, we justify the uniqueness of u (the uniqueness of b(u) is well-known) and prove the continuous dependence in L ¹ for the case of strongly convergent finite energy data. We also prove convergence of the ${\varepsilon}$ -discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux F, we show that the problem admits a maximal and a minimal solution.     

Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces

We consider the first boundary value problem and the oblique derivative problem for a linear second-order parabolic equation in noncylindrical not necessarily bounded domains with nonsmooth (with respect to t) and noncompact lateral boundary under the assumption that the right-hand side and the lower-order coefficients of the equation may have certain growth when approaching the parabolic boundary of the domain and all coefficients are locally Hölder with given characteristics of the Hölder property. We construct a smoothness scale of solutions of these boundary value problems in Hölder spaces of functions that admit growth of higher derivatives near the parabolic boundary of the domain.     

Strongly nonlinear elliptic boundary value problems in Musielak–Orlicz spaces

Monatshefte für Mathematik - In this paper we prove an existence result of solutions for some strongly nonlinear elliptic problems with lower order term and $$L^1$$ -data in...     

Regularity of solutions of boundary value problems for parabolic equations of arbitrary order in weighted Hölder spaces

We consider initial-boundary value problems for a uniformly parabolic equation of arbitrary order 2m in a noncylindrical domain whose lateral boundary is nonsmooth with respect to t. We assume that the lower-order coefficients and the right-hand side of the equation, generally speaking, grow to infinity no more rapidly than some power function when approaching the parabolic boundary of the domain, all coefficients of the equation are locally Hölder, and their Hölder constants can grow near that boundary. We construct a smoothness scale of solutions of such problems in weighted Hölder classes of functions whose higher derivatives may grow when approaching the parabolic boundary of the domain.     

Global existence of solutions to a parabolic–elliptic chemotaxis system with critical degenerate diffusion

This paper is devoted to the analysis of nonnegative solutions for a degenerate parabolic–elliptic Patlak–Keller–Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, thereby completing previous results on finite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.     

Optimal control of a nonlinear parabolic–elliptic system

In this paper, we shall study the problem of optimal control of the parabolic–elliptic system

_ut+_(f(t,x,u))x+g(t,x,u)+_Px−_(a(t,x)ux)x=_f0+B^∗ν$u_t+{\left(f(t,x,u)\right)}_x+g(t,x,u)+P_x-{\left(a(t,x)u_x\right)}_x=f_0+B^{\ast }\nu$