This paper is the parabolic counterpart of previous ones about elliptic operators in unbounded domains. Maximum principles for second-order linear parabolic equations are established showing a variant of the ABP-Krylov-Tso estimate, based on the extension of a technique introduced by Cabré, which in turn makes use of a lower bound for super-solutions due to Krylov and Safonov. The results imply the uniqueness for the Cauchy-Dirichlet problem in a large class of innite cylindrical and non-cylindrical domains.

We consider the equation u_t = Tr[B(x, t, Du, Φu)D²u] + F(x, t, u, Du, Φu, ψu) where Φ and ψ are vector-valued mappings. We obtain the existence and uniqueness of classical solution to the equation for a ∈- periodic initial data. The problem is naturally arisen from image denoising.

In this paper, we consider the Cauchy problem for some dispersive equations. By means of nonlinear estimate in Besov spaces and fixed point theory, we prove the global well-posedness of the above problem. What's more, we improve the scattering result obtained in [1].

We study the exponential stability of traveling wave solutions of non-linear systems of integral differential equations arising from nonlinear, nonlocal, synaptically coupled, excitatory-inhibitory neuronal networks. We have proved that exponential stability of traveling waves is equivalent to linear stability. Moreover, if the real parts of nonzero spectrum of an associated linear differential operator have a uniform negative upper bound, namely, max{Reλ: λ ∈ σ(L), λ ≠ 0} ≤ -D, for some positive constant D, and λ = 0 is an algebraically simple eigenvalue of L, then the linear stability follows, where L is the linear differential operator obtained by linearizing the nonlinear system about its traveling wave and σ(L) denotes the spectrum of L. The main aim of this paper is to construct complex analytic functions (also called eigenvalue or Evans functions) for exploring eigenvalues of linear differential operators to study the exponential stability of traveling waves. The zeros of the eigenvalue functions coincide with the eigenvalues of L. When studying multipulse solutions, some components of the traveling waves cross their thresholds for many times. These crossings cause great difficulty in the construction of the eigenvalue functions. In particular, we have to solve an over-determined system to construct the eigenvalue functions. By investigating asymptotic behaviors as z → -∞ of candidates for eigenfunctions, we find a way to construct the eigenvalue functions. By analyzing the zeros of the eigenvalue functions, we can establish the exponential stability of traveling waves arising from neuronal networks.

In this paper, we consider the existence and nonexistence of global solutions to the semilinear heat equation u_t - Δu = u^p with Neumann boundary value \frac{∂u}{∂ν} = 0 on some unbounded domains, where p > 1, ν is the outward normal vector on boundary ∂Ω. We prove that there exists a critical exponent p_c = p_c(Ω) > 1 such that if p ∈ (1, p_c], for nonnegative and nontrivial initial data, all positive solutions blow up in finite time; if p > p_c, for suitably small nonnegative initial data, there exists a global positive solution.

In this paper we deal with the self-similar singular solution of the p-Laplacian evolution equation u_t = div(|∇|^{p-2}∇u) - |∇u|^q for p > 2 and q > 1 in R^n × (0, ∞). We prove that when p > q + n/(n + 1) there exist self-similar singular solutions, while p ≤ q+n/(n+1) there is no any self-similar singular solution. In case of existence, the self-similar singular solutions are the self-similar very singular solutions, which have compact support. Moreover, the interface relation is obtained.