This paper studies V-form traveling fronts to a bistable reaction–diffusion equation. V-form traveling fronts have been studied so far for many reaction–diffusion equations. The proof of its existence is often based on the multi-scale method. In this paper we present a new simple proof for the existence of V-form traveling fronts based on estimates by inscribed balls.
We consider the Cauchy problem for a quadratic derivative nonlinear Schrödinger equation on the line. In particular, we give a condition that the flow map fails to be uniformly continuous on small balls in the L^2 -based Sobolev space H^s(ℝ) for s>3/2 . For the proof, we combine the method of Koch–Tzvetkov [16] with the gauge transformation introduced by Hayashi [6] and Hayashi–Ozawa [7]. The key point in the proof is to construct solutions in the space H^s(ℝ) with bounded primitives which was introduced by Ozawa [23].
In this work, we study a non-conservative quasilinear first order system in space dimension d. As global smooth solutions do not exist even with smooth initial data, the solution must be understood in a weak sense. For space dimension d=1 , there is a well developed theory based on the definition of non-conservative product [1]. For d>1 , this analysis does not work because of the lack of estimates in the space of BV functions. First, we prove the existence of weak asymptotic solutions, with general initial conditions. Further, we present the explicit formula for the gradient type solutions and the plane wave solutions. In this analysis, we use the viscous approximation of the system with variable viscosity. We also study boundary value problems for space dimension d=1 and get the explicit formula for the weak solution. The boundary condition here must be understood in a weak sense. The theme of this paper is to study non-conservative quasilinear first order systems of partial differential equations in the d-space dimension. This includes the analysis of partial differential equations that model the evolution of density inhomogeneities in matter in the universe.
This paper focuses on an optimal control problem for a system of five nonlinear parabolic equations derived from population dynamics. We specifically analyze a diffusive SVEIR epidemic model, which incorporates Susceptible, Vaccinated, Exposed, Infected, and Recovered individuals, with some class of nonlinear incidences. We propose two control strategies aimed at reducing the number of susceptible, vaccinated, and infectious individuals, while minimizing costs. Firstly, we establish the existence, uniqueness, and certain estimates for the strong solution of the state system, utilizing the semigroup theory combined with the truncation technique. Furthermore, using the minimizing sequence technique, we prove the existence of an optimal control. Finally, we derive the first-order optimality conditions and characterize the optimal control by expressing it in terms of the state functions and adjoint functions. The optimality system is formulated and subsequently solved numerically using an algorithm that employs forward and backward difference approximations.
This paper is concerned with the existence and symmetry breaking properties of vectorial ground states for the following fractional Hartree–Fock system with a general potential in ℝ^2 : {[ (-Δ )^su+u+l(x)ϕ ^t_l,(u,v)u=( |u|^2p+|v|^p+1|u|^p-1) u in ℝ^2,; (-Δ )^sv+v+l(x)ϕ ^t_l,(u,v)v=( |v|^2p+|u|^p+1|v|^p-1) v in ℝ^2 , ]. where 1/2
