Weak solutions of hyperbolic-parabolic Volterra equations

Gripenberg, Gustaf

doi:10.1090/s0002-9947-1994-1216335-0

Cited by 9 publications

(6 citation statements)

References 28 publications

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“…Assuming that the stress depends nonlinearly on y x , that is, τ = σ ( y x ), where σ is the first Piola–Kirchhoff stress component in one space dimension (see (2.2) for the definition), we obtain the nonlinear wave equation ρR ytt=σfalse(yxfalse)x,2emfalse(x,tfalse)∈normalΩ×false(0,normal∞false), where σ = W ′ (see (2.2) for the general definition) stands for the non-monotone stress as in figure 2. By the results of MacCamy & Mizel [27], we know that global solutions for equation (3.2), even for smooth initial data, do not exist in general due to the fact that second derivatives of the solutions may become infinite in finite time (see also the discussion in [28] for a hyperbolic-parabolic formulation related to Volterra equations and [29] for a discussion in three-space dimension). In particular, as Pego [30] explains, when y x is in the ranges where σ is decreasing, the equation becomes elliptic and this makes the initial value problem ill-posed.…”

Section: Governing Equationsmentioning

confidence: 99%

Nonlinear viscoelasticity of strain rate type: an overview

Şengül

2021

Proc. R. Soc. A.

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There are some materials in nature that experience deformations that are not elastic. Viscoelastic materials are some of them. We come across many such materials in our daily lives through a number of interesting applications in engineering, material science and medicine. This article concerns itself with modelling of the nonlinear response of a class of viscoelastic solids. In particular, nonlinear viscoelasticity of strain rate type, which can be described by a constitutive relation for the stress function depending not only on the strain but also on the strain rate, is considered. This particular case is not only favourable from a mathematical analysis point of view but also due to experimental observations, knowledge of the strain rate sensitivity of viscoelastic properties is crucial for accurate predictions of the mechanical behaviour of solids in different areas of applications. First, a brief introduction of some basic terminology and preliminaries, including kinematics, material frame-indifference and thermodynamics, is given. Then, considering the governing equations with constitutive relationships between the stress and the strain for the modelling of nonlinear viscoelasticity of strain rate type, the most general model of interest is obtained. Then, the long-term behaviour of solutions is discussed. Finally, some applications of the model are presented.

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Section: Governing Equationsmentioning

confidence: 99%

Nonlinear viscoelasticity of strain rate type: an overview

Şengül

2021

Proc. R. Soc. A.

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“…The FPDE (1.3)-(1.4) interpolates two PDEs (see e.g. [9]), namely semilinear wave (α = 2) and heat (α = 1) equations, which have been widely investigated in the last years. These PDEs present many differences in the theory of existence and asymptotic behavior of solutions in scaling invariant spaces (critical spaces).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Existence and symmetries of solutions in Besov–Morrey spaces for a semilinear heat-wave type equation

Almeida

Precioso

2015

Journal of Mathematical Analysis and Applications

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This paper considers a semilinear integro-differential equation of Volterra type which interpolates semilinear heat and wave equations. Global existence of solutions is showed in spaces of Besov type based in Morrey spaces, namely BesovMorrey spaces. Our initial data is larger than the previous works and our results provide a maximal existence class for semilinear interpolated heat-wave equation. Some symmetries, self-similarity and asymptotic behavior of solutions are also investigated in the framework of Besov-Morrey spaces.

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“…For ao 4 3 this task is essentially solved (see [10]). By different methods, the existence, but not the uniqueness, of a solution u satisfying uAW 1;N loc ðR þ ; L 2 ð0; 1ÞÞ-L 2 loc ðR þ ; W 2;2 0 ð0; 1ÞÞ was proved in [12], for the range aA½ 4 3 ; 3 2 : For 3 2 oao2; only existence of global weak solutions has been proved [11]. We do however conjecture that unique smooth, global solutions do exist for the entire range aAð0; 2Þ:…”

Section: An Examplementioning

confidence: 99%

Quasilinear evolutionary equations and continuous interpolation spaces

Clément

Londen²,

Simonett

2004

Journal of Differential Equations

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In this paper we analyze the abstract parabolic evolutionary equations D a t ðu À xÞ þ AðuÞu ¼ f ðuÞ þ hðtÞ; uð0Þ ¼ x;in continuous interpolation spaces allowing a singularity as tk0: Here D a t denotes the timederivative of order aAð0; 2Þ: We first give a treatment of fractional derivatives in the spaces L p ðð0; TÞ; X Þ and then consider these derivatives in spaces of continuous functions having (at most) a prescribed singularity as tk0: The corresponding trace spaces are characterized and the dependence on a is demonstrated. Via maximal regularity results on the linear equationwe arrive at results on existence, uniqueness and continuation on the quasilinear equation.Finally, an example is presented. r

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Weak solutions of hyperbolic-parabolic Volterra equations

Abstract: Abstract. The existence of a global weak solution, satisfying certain a priori L°°-bounds, of the equation ut(t, x) = /0' k(t -s)(o(ux))x{s, x)ds + f{t, x) is established. The kernel k is locally integrable and log-convex, and a1 has only one local minimum which is positive.

Cited by 9 publications

References 28 publications

Nonlinear viscoelasticity of strain rate type: an overview

Nonlinear viscoelasticity of strain rate type: an overview

Existence and symmetries of solutions in Besov–Morrey spaces for a semilinear heat-wave type equation

Quasilinear evolutionary equations and continuous interpolation spaces

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