Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting

The local solution of a parabolic-elliptic equation with a nonlinear Neumann boundary condition

Abstract. We investigate a parabolic-elliptic problem, where the time derivative is multiplied by a coefficient which may vanish on time-dependent spatial subdomains. The linear equation is supplemented by a nonlinear Neumann boundary condition −∂u/∂νA = g(·, ·, u) with a locally defined, Lr-bounded function g(t, ·, ξ). We prove the existence of a local weak solution to the problem by means of ...

A Spectral Method for an Elliptic Equation with a Nonlinear Neumann Boundary Condition

Abstract. Let Ω be an open, simply connected, and bounded region in Rd, d ≥ 2, that is diffeomorphic to Bd. Consider solving −∆u+ γu = 0 over Ω with the Neumann boundary condition ∂u ∂n = b (·, u). The function b is a nonlinear function of u. The problem is reformulated in a weak form, and then a spectral Galerkin method is used to create a sequence of finite dimensional nonlinear problems. An ...

Existence of Nontrivial Solution for a Nonlocal Elliptic Equation with Nonlinear Boundary Condition

A Fourth Order Elliptic Equation with Nonlinear Boundary Conditions

In this paper we study the existence of infinitely many nontrivial solutions of the following problem, −∆2u = u in Ω, − ∂∆u ∂ν = f(x, u) on ∂Ω, and either ∂u ∂ν = 0 or ∆u = 0 on ∂Ω. We assume that f(x, u) is superlinear and either subcritical or a sublinear perturbation of the critical case. For the proof in the critical case we apply the concentration compactness method.

Porous medium equation with absorption and a nonlinear boundary condition

In this paper we study a porous medium equation with a nonlinear absorption term and a nonlinear boundary condition. We prove existence of weak solutions and also we establish some uniqueness and non uniqueness results for certain range of the parameters that appear in the problem. Finally we deal with the existence of global solutions in time or blow-up. We find in which region of parameters t...