Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

We study the nonlinear Neumann boundary value problem for semilinear heat equation ∂ t u − Δ u = λ | u | p , t > 0 , x ∈ R + n , u ( 0 , x ) = ε u 0 ( x ) , x ∈ R + n , − ∂ x u ( t , x ′ , 0 ) = γ | u | q ( t , x ′ , 0 ) , t > 0 , x ′ ∈ R n − 1 where p = 1 + 2 n , q = 1 + 1 n and ε > 0 is small enough. We investigate the life span of solutions for λ , γ > 0. Also we study the global in time existence and large time asymptotic behavior of solutions in the case of λ , γ < 0 and ∫ R + n u 0 ( x ) d x > 0.

An Exterior Neumann Boundary-Value Problem for the Div-Curl System and Applications

We investigate a generalization of the equation curlw→=g→ to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of Rn using classical operators from Clifford analysis. In the physically significant case n=3, two explicit solutions to the div-curl system in exterior domains of R3 are obtained following different constructions of hyper-conjugate harmonic pairs. One of the constructions hinges on the use of a radial integral operator introduced recently in the literature. An exterior Neumann boundary-value problem is considered for the div-curl system. That system is conveniently reduced to a Neumann boundary-value problem for the Laplace equation in exterior domains. Some results on its uniqueness and regularity are derived. Finally, some applications to the construction of solutions of the inhomogeneous Lamé–Navier equation in bounded and unbounded domains are discussed.

How far does logistic dampening influence the global solvability of a high-dimensional chemotaxis system?

This paper deals with the homogeneous Neumann boundary value problem for chemotaxis system $$\begin{aligned} \textstyle\begin{cases} u_{t} = \Delta u - \nabla \cdot (u\nabla v)+\kappa u-\mu u^{\alpha }, & x\in \Omega, t>0, \\ v_{t} = \Delta v - uv, & x\in \Omega, t>0, \end{cases}\displaystyle \end{aligned}$$ { u t = Δ u − ∇ ⋅ ( u ∇ v ) + κ u − μ u α , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}(N\geq 2)$ Ω ⊂ R N ( N ≥ 2 ) , where $\alpha >1$ α > 1 and $\kappa \in \mathbb{R},\mu >0$ κ ∈ R , μ > 0 for suitably regular positive initial data.When $\alpha \ge 2$ α ≥ 2 , it has been proved in the existing literature that, for any $\mu >0$ μ > 0 , there exists a weak solution to this system. We shall concentrate on the weaker degradation case: $\alpha <2$ α < 2 . It will be shown that when $N<6$ N < 6 , any sublinear degradation is enough to guarantee the global existence of weak solutions. In the case of $N\geq 6$ N ≥ 6 , global solvability can be proved whenever $\alpha >\frac{4}{3}$ α > 4 3 . It is interesting to see that once the space dimension $N\ge 6$ N ≥ 6 , the qualified value of α no longer changes with the increase of N.

An Optimal Bound for Nonlinear Eigenvalues and Torsional Rigidity on Domains with Holes

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.