Nonlinear Differential Equations of Monotone Types in Banach Spaces

Abstract. Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.

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“…In addition, it is clear that the operator T 3 + I is surjective. Therefore, T 3 is maximal monotone (see Theorem 2.2 in [1]). …”

Section: 1mentioning

confidence: 96%

“…To justify the surjectivity of T , it is sufficient to show that the operator T is coercive and maximal monotone (Corollary 2.2 in [1]). …”

Section: 1mentioning

confidence: 99%

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Guo

Rammaha

Sakuntasathien

et al. 2014

Journal of Differential Equations

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“…For other definitions, results and details regarding these topics, we refer the reader to Barbu [3] and Vrabie [21]. For more details concerning functional differential equations with delay we refer to Hale [16].…”

Section: Preliminariesmentioning

confidence: 99%

A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

Burlică¹,

Roșu²

2015

Annals of the Alexandru Ioan Cuza University - Mathematics

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“…Even if the reflexivity of X is dropped, i.e., (3.1) might not have a strong solution, the continuous function t → e tg η is still well defined and one then refers to it as a mild solution of (3.1). Surveys of the nonlinear semigroup theory can be found in [4,6,10].…”

Section: Mild Solutions and The Implicit Euler Schemementioning

confidence: 99%

Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations

Hansen

Stillfjord

2013

Math. Comp.

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Abstract. We present a convergence analysis for the implicit-explicit (IMEX) Euler discretization of nonlinear evolution equations. The governing vector field of such an equation is assumed to be the sum of an unbounded dissipative operator and a Lipschitz continuous perturbation. By employing the theory of dissipative operators on Banach spaces, we prove that the IMEX Euler and the implicit Euler schemes have the same convergence order, i.e., between one half and one depending on the initial values and the vector fields. Concrete applications include the discretization of diffusion-reaction systems, with fully nonlinear and degenerate diffusion terms. The convergence and efficiency of the IMEX Euler scheme are also illustrated by a set of numerical experiments.

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Nonlinear Differential Equations of Monotone Types in Banach Spaces

Cited by 663 publications

References 0 publications

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations

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