Semilinear Cauchy Problem Research Articles

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

Up to now, studies on the semi-linear Cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global Lipschitz function. The current paper is the first investigation to not only the more general but also the more practical case of interest when the source term is only a local Lipschitz function. In such a situation, the methods of solution from the previous studies with a global Lipschitz source term are not directly applicable and therefore, novel ideas and techniques need to be developed to tackle the local Lipschitz nonlinearity. This locally Lipschitz source arises in many applications of great physical interest governed by, for example, the sine-Gordon, Lane–Emden, Allen–Cahn and Liouville equations. The inverse problem is severely ill-posed in the sense of Hadamard by violating the continuous dependence upon the input Cauchy data. Therefore, in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method. Under some priori assumptions on the exact solution, we prove and obtain rigorously convergence estimates.

Nonlocal Cauchy problems close to an asymptotically stable equilibrium point

In this work we investigate the existence of solutions for semilinear Cauchy problems with nonlocal initial conditions in the neighborhood of an asymptotically stable equilibrium point of the evolution equation. Using Granas' continuation principle for contractive maps and the qualitative theory of differential equations in Banach spaces, under mild assumptions, we prove the existence of a unique solution. We also show that the main abstract result can be applied to nonlocal initial boundary value problems for reaction–diffusion equations with non-convex nonlinearities.

Reaction-Advection-Diffusion Equations with Space Fractional Derivatives and Variable Coefficients on Infinite Domain

Abstract In this paper we consider a space fractional reaction-advection-diffusion equation which is actually a semi-linear Cauchy problem with a spatial fractional derivative operator of order α, 0 < α < 2. We establish and prove assertions concerning the existence and uniqueness of solution within certain Colombeau space. The proofs are given in the case when the left Liouville fractional derivative is involved, but the results are also valid in the case of the right Liouville fractional derivative as well as for the Riesz fractional derivative. The solutions are obtained by using the theory of generalized uniformly continuous semigroups of operators. We have also proved that the non-regularized and corresponding regularized operators, as well as solutions of non-regularized and corresponding regularized equations are L2−associated.

Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis.

In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate apbq (where a and b are the concentrations of A and B, respectively, with 0<p,q<1) and well-posedness for this problem has been lacking up to the present.

Large Time Behavior of Solutions to SemiLinear Equations with Quadratic Growth in the Gradient

This paper studies the large time behavior of solutions to semilinear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space. Two types of large time behavior are obtained: (i) pointwise convergence of the solution and its gradient and (ii) convergence of solutions to associated backward stochastic differential equations. When the state space is $\mathbb{R}^d$ or the space of positive definite matrices, both types of convergence are obtained under growth conditions on coefficients. These large time convergence results have direct applications in risk-sensitive control and long-term portfolio choice problems.

A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton---Jacobi---Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.

Normal forms for semilinear equations with non-dense domain with applications to age structured models

Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator and present procedures to compute the Taylor expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model.

On the stability of the positive radial steady states for a semilinear Cauchy problem

This paper is a contribution to the analysis of the Cauchy problem (0.1){∂u∂t=Δu+∑i=1kci|x|liupiinRn×(0,T),u(x,0)=φ(x)inRn, with initial function φ≢0. The stability and asymptotic stability of the positive radial steady states, which are positive solutions of (0.2)Δu+∑i=1kci|x|liupi=0, have been discussed with φ in weighted Banach spaces.

The Stability of the Positive Radical Steady States for a Nonhomogeneous Semilinear Cauchy Problem

The Stability of the Positive Radical Steady States for a Nonhomogeneous Semilinear Cauchy Problem

The global existence of mild solutions for semilinear fractional Cauchy problems in the α-norm

The global existence of mild solutions for semilinear fractional Cauchy problems in the α-norm

Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients

Uniqueness and nonuniqueness of solutions to the first initial-boundary value problem for degenerate semilinear parabolic equations, with possibly unbounded coefficients, are studied. Sub- and supersolutions of suitable auxiliary problems, such as the first exit time problem, are used to determine on which part of the boundary Dirichlet data must be given. As an application of the general results, we study uniqueness and nonuniqueness of bounded solutions to a semilinear Cauchy problem in the hyperbolic space $\mathbb{H}^n$ using the Poincare model.

Long time asymptotics of sub-threshold solutions of a semilinear Cauchy problem

We show that the solution of the semilinear heat equation u t = uxx +u p (with x ∈ R, p > 3, and nonnegative Cauchy data) behaves for large t like the solution of the corresponding linear problem plus a small correction of order t −1/2−c ,w herec := 1/2, if p 4, and c = (p −3)/2, if 3< p < 4. The result is known in special cases like small initial data. We prove it here for positive sub-threshold initial data satisfying some assumptions. Part of our results are contained in the recent work (10), but the motivation of this paper is to provide a new method leading to somewhat more general space-time estimates. 1. Main result We consider the long time asymptotical behaviour of the solution of the classi- cal semilinear diffusion equation and the associated Cauchy problem for the unknown function u = u(x,t), x ∈ R, t 0:

S -asymptotically ω-periodic solutions for semilinear Volterra equations

We study S-asymptotically ω-periodic mild solutions of the semilinear Volterra equation u′(t)=(a* Au)(t)+f(t, u(t)), considered in a Banach space X, where A is the generator of an (exponentially) stable resolvent family. In particular, we extend the recent results for semilinear fractional integro-differential equations considered in (Appl. Math. Lett. 2009; 22:865–870) and for semilinear Cauchy problems of first order given in (J. Math. Anal. Appl. 2008; 343(2): 1119–1130). Applications to integral equations arising in viscoelasticity theory are shown. Copyright © 2010 John Wiley & Sons, Ltd.

On semilinear Cauchy problems with non-dense domain

We are interested in studying semilinear Cauchy problems in which the closed linear operator is not Hille-Yosida and its domain is not densely defined. Using integrated semigroup theory, we study the positivity of solutions to the semilinear problem, the Lipschitz perturbation of the problem, differentiability of the solutions with respect to the state variable, time differentiability of the solutions, and the stability of equilibria. The obtained results can be used to study several types of differential equations, including delay differential equations, age-structure models in population dynamics, and evolution equations with nonlinear boundary conditions.

Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain

In this work we study the existence and uniqueness of compact almost automorphic solutions to a first-order differential equation with a linear part dominated by a Hille–Yosida type operator with non dense domain.

Local mild solutions and impulsive mild solutions for semilinear Cauchy problems involving lower Scorza-Dragoni multifunctions

In this note we investigate in Banach spaces the existence of mild solutions for initial problems, also in presence of impulses, governed by semilinear differential inclusions where the non-linear part is a Scorza-Dragoni multifunction. All the results are obtained via a generalization of {\it Artstein-Prikry selection theorem} that we obtain in the first part of the paper.

On the Stability of the Positive Radial Steady States for a Semilinear Cauchy Problem Involving Critical Exponents

This article is contributed to the Cauchy problem { ∂ u ∂ t = Δ u + K ( | x | ) u p in R n × ( 0 , T ) , u ( x , 0 ) = ϕ ( x ) in R n ; with initial function φ notequiv; 0. The stability of positive radial steady state, which are positive solutions of Δ u + K ( | x | ) u p = 0 , is obtained when p is critical for general K(|x|).

On the semistability of the minimal positive steady state for a nonhomogeneous semilinear Cauchy problem

This paper is contributed to the study of the Cauchy problem \[ \left\{ \begin{array}{ll} u_{t}=\Delta u+K(|x|)u^{p}+\mu f(|x|) \mbox{in} R^{n}\times (0,T), & u(x,0)=\varphi (x) \mbox{in} R^{n}, & u(x,0)=\varphi (x) \mbox{in} R^{n}, & \end{array}% \right. \]% with non-negative initial function $\varphi \not\equiv 0$. We will study the asymptotic behavior and the semistability of the minimal positive steady state. In addition, we will prove that all slow decay positive steady states are stable and weakly asymptotically stable in some weighted $L^{\infty }$ norms.

Semilinear Cauchy Problems with Almost Sectorial Operators

Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem.

Morawetz-Strichartz estimates for spherically symmetric solutions to wave equations and applications to semilinear Cauchy problems

We prove some space-time estimates with homogeneous weight functions for the wave equation with radially symmetric data in order to explain a connection between the Morawetz-type space-time estimate and the Strichartz-type space-time estimate. As an application of such linear estimates, we give a new proof of global existence of small radial solutions to semilinear wave equations. Local existence of radial solutions is also studied for low-regularity data.