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Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity

Cordeiro Sebastião, Raposo Carlos, Ferreira Jorge, Rocha Daniel, Shahrouzi Mohammad

Opuscula Mathematica

2024

Vol. 44, no. 1

19--47

This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma.

Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain

Czernous W.

Opuscula Mathematica

2014

Vol. 34, no. 2

291--310

We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain Ω with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of ∂Ω with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on (0,c)×Ω to the initial boundary value problem, for small c. Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.

Classical solutions of initial problems for quasilinear partial functional differential equations of the first order

Czernous W.

Opuscula Mathematica

2006

Vol. 26, no. 1

13-29

We consider the initial problem for a quasilinear partial functional differential equation of the first order [formula], z(t, x) = varphi(t, x) ((t, x) ∈ [-h0, 0] x Rn) where z(t, x) : [-h0, 0] x [-h, h] → R is a function defined by z(t, x) (τ, ξ) = z(t + τ, + ξ) for (τ, ξ) ∈ [-h0, 0] x [-h, h]. Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition.

Local existence of solutions of a free boundary problem for equations of compressible viscous heat-conducting capillary fluids

Zadrzyńska E., Zajączkowski W. M.

Journal of Applied Analysis

2000

Vol. 6, nr 2

227-250

In the paper the motion of a viscous compressible heat conducting capillary fluid in a domain bounded by a free surface is considered. We prove the local existence and uniqueness of a solution to a problem describing such a motion in anisotropic Sobolev-Slobodetskii spaces. This solution is such that the velocity and temperature belong to (wzór), and density to (wzór).