Global existence and uniqueness for viscoelastic equations with nonstandard growth conditions

Abita Rahmoune

Abstract


This paper is devoted to the study of generalized viscoelastic nonlinear equations with Dirichlet-Neumann boundary conditions. We establish the local and uniqueness of weak solutions results in Sobolev spaces with variable exponents. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space.


Keywords


Viscoelastic equation, Global Existence, Nonlinear Dissipation, Energy estimates.

Full Text:

PDF

References


Abita, R.,

Semilinear hyperbolic boundary value

problem associated to the nonlinear generalized viscoelastic equations,

Acta Mathematica Vietnamica, {bf 43}(2018), 219-238.

Abita, R.,

Existence and asymptotic stability for the

semilinear wave equation with variable-exponent nonlinearities,

J. Math. Phys., {bf 60}(2019), 122701.

Abita, R.,

Bounds for below-up time in a nonlinear

generalized heat equation,

Appl Anal., (2020), 1871-1879.

Abita, R.,

Lower and upper bounds for the blow-up time

to a viscoelastic Petrovsky wave equation with variable sources and memory

term, Appl Anal., (2022), 1-29.

Andradea, D., Jorge Silvab, M.A., Mac, T.F.,

Exponential stability for a plate equation with $p$-laplacian and

memory terms,

Math. Methods Appl. Sci., {bf 35}(2012), 417-426.

Ayang, Z.,

Global existence, asymptotic behavior and

blow-up of solutions for a class of nonlinear wave equations with

dissipative term, J. Differential Equations, {bf 187}(2003), 520-540.

Ayang, Z., Baoxia, J.,

Global attractor for a class of Kirchhoff models,

J. Math. Phys., {bf 50}(2010), 29pp.

Cavalcanti, M.M., Oquendo, H.P.,

Frictional versus viscoelastic damping in a semilinear wave equation,

SIAM J. Control Optim., {bf 42}(2003), 1310-1324.

Dafermos, C.M.,

Asymptotic stability in viscoelasticity,

Arch. Rational Mech. Anal., {bf 37}(1970), 297-208.

Dafermos, C.M., Nohel, J.A.,

Energy methods for

nonlinear hyperbolic volterra integro-differential equations,

Comm. Partial Differential Equations, {bf 4}(1979), 219-278.

Diening, L., Histo, P., Harjulehto, P., Ru{u}zicka, M.,

Lebesgue and Sobolev Spaces with Variable Exponents,

vol. 2017, in: Springer Lecture Notes, Springer-Verlag, Berlin, 2011.

Diening, L., Ru{u}zicka, M.,

Calderon Zygmund operators on generalized Lebesgue spaces $L^{p(x)}(Omega )$ and

problems related to fluid dynamics,

Preprint Mathematische Fakult"{a}t,

Albert-Ludwigs-Universit"{a}t Freiburg, {bf 120}(2002), 197-220.

Fan, X., Shen, J., Zhao, D.,

Sobolev embedding theorems for spaces $W^{k,p(x)}(Omega )$,

J. Math. Anal. Appl., {bf 262}(2001), 749-760.

Fu, Y.,

The existence of solutions for elliptic systems

with nonuniform growth, Studia Math., {bf 151}(2002), 227-246.

Kov'{r}cik, O., R'{a}kosnik, J.,

On spaces $L^{p(x)}(Omega )$ and $W^{1,p(x)}(Omega )$,

Czechoslovak Math. J.,

{bf 41}(1991), 592-618.

Lions, J.L.,

Quelques M'{e}thodes de R'{e}solution des Drobl`{e}mes aux Limites Non Lin'{e}aires,

Dunod, Paris, 1966.

Ma, T.F., Soriano, J.A.,

On weak solutions for an evolution equation with exponential nonlinearities,

Nonlinear Anal., {bf 37}(1999), 1029-1038.

Rivera JE, M.,

Asymptotic behaviour in linear viscoelasticity,

Quart. Appl. Math., {bf 52}(1994), 628-648.

Rivera JE, M., Andrade, D.,

Exponential decay of non-linear wave equation with a viscoelastic boundary condition,

Math. Methods Appl. Sci., {bf 23}(2000), 41-61.




DOI: http://dx.doi.org/10.24193/subbmath.2024.2.12

Refbacks

  • There are currently no refbacks.