In this paper, we consider the system ut = ∆u, vt = ∆v x ∈ R+ , t > 0, − ∂u ∂x1 = v, − ∂v ∂x1 = u x1 = 0, t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x) x ∈ R+ , where R+ = {(x1, x′) | x′ ∈ RN−1, x1 > 0}, p, q > 0, and u0, v0 nonnegative. We prove that if pq ≤ 1 every nonnegative solution is global. When pq > 1 we let α = 1 2 p+1 pq−1 , β = 1 2 q+1 pq−1 . We show that if max(α, β) ≥ N 2 , all nontrivi...

in this article, we develop the distributed order fractional hybrid differential equations (dofhdes) with linear perturbations involving the fractional riemann-liouville derivative of order $0 < q < 1$ with respect to a nonnegative density function. furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved under the mixed $varphi$-lipschit...

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction...

In this paper, we study the existence of linear quadratic Gaussian (LQG)–balanced realizations for discrete-time infinite-dimensional systems. LQG-balanced realizations are those for which the smallest nonnegative self-adjoint solutions of the control and filter Riccati equations are equal. We show that the control (filter) Riccati equation has a nonnegative self-adjoint solution if and only if...

This work studies nonnegative solutions for the Cauchy, Neumann, and Dirichlet problems of the logistic type equation ut = ∆u+ μu p − a(x)u with p, q > 1, μ > 0. The finite time blowup results for nonnegative solutions under various restrictions on a(x), p, q, μ are presented. Applying the results allows one to construct some reaction diffusion systems with

This paper deals with a degenerate parabolic equation vt = ∆v + av1 ∥v∥1 α1 subject to homogeneous Dirichlet condition. The local existence of a nonnegative weak solution is given. The blow-up and global existence conditions of nonnegative solutions are obtained. Moreover, we establish the precise blow-up rate estimates for all the blow-up solutions.

We consider a nonsymmetric algebraic matrix Riccati equation arising from transport theory. The nonnegative solutions of the equation can be explicitly constructed via the inversion formula of a Cauchy matrix. An error analysis and numerical results are given. We also show a comparison theorem of the nonnegative solutions.

In this paper we propose a structure-preserving doubling algorithm (SDA) for computing the minimal nonnegative solutions to the nonsymmetric algebraic Riccati equation (NARE) based on the techniques developed in the symmetric cases. This method allows the simultaneous approximation of the minimal nonnegative solutions of the NARE and its dual equation, only requires the solutions of two linear ...