The Cauchy problem for fractional $m$-evolution models with memory: Well-Posedness and blow-up time estimates
Abstract
In this paper, we investigate the well-posedness and establish lower and upper bounds for the blow-up time of solutions to a class of fractional Laplacian equations. The governing model includes a nonlinear source term and dissipative effects with variable exponents. The equation features a wave-like structure with a fractional diffusion term, a memory term involving a convolution with a relaxation kernel, a nonlinear damping term with a variable exponent, and a source term with another variable exponent. The kernel is assumed to be smooth, non-increasing, and satisfies a specific smallness condition on its total integral. We prove the existence of a solution and then derive lower and upper bounds for the blow-up time, which depend on the fractional exponent, the variable growth exponents in the damping and source terms, and the properties of the relaxation kernel.
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References
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Nonlinear Analysis 112 (2015) 129--146.
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\bibitem{11} Piskin, E. Blow up of solutions for a nonlinear viscoelastic wave equations with variable exponents. Middle East J. Sci. 2019, 5,
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\bibitem{park} Park S-H. General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities. Bound Value Probl.
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Nonlinear Analysis 112 (2015) 129--146.
\bibitem{Mess1} Messaoudi SA, Talahmeh AA, Al-Smail JH. Nonlinear damped wave equation: existence and blow-up. Comput Math Appl. 2017;74:3024-3041.
\bibitem{Mess2} Messaoudi SA, Talahmeh AA. Blow-up in solutions of a quasilinear wave equation with variable-exponent nonlinearities. Math Meth
Appl Sci. 2017;40:6976-6986.
\bibitem{Ra} A. Rahmoune, Lower and upper bounds for the blow-up time to a viscoelastic Petrovsky wave equation with variable sources and memory term,
Appl. Anal., 102(12) (2023) 3503--3531.
\bibitem{Messaoudi} Ilyes Lacheheba and Salim A. MessaoudiGeneral decay of the Cauchy problem for $\alpha $-evolution models with $\beta $-memory term.
Applicable Analysis, 104(8) (2025), 1473--1486.
\bibitem{Orlicz} Orlicz W. Uber konjugierte Exponentenfolgen. Studia Math., 3:200--212, 1931.
\bibitem{i1} Aboulaich R, Meskine D, Souissi A. New diffusion models in image processing. Comput Math Appl. 2008;56(4):874-882.
\bibitem{i3} Lian S, GaoW, Cao C, Yuan H. Study of the solutions to a model porous medium equation with variable exponent of nonlinearity. JMath Anal
Appl. 2008, 342(1):27-38.
\bibitem{1} E. Acerbi, G. Mingione, Regularity results for stationary eletrorheological fluids. Arch. Ration. Mech. Anal. \textbf{164}(2002),
213-259.
\bibitem{2} S.N. Antonsev, S.I. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math. \textbf{%
234}(2010), 2633-2645.
\bibitem{3} L. Diening, P. Harjulehto, P. H\"{a}st\"{o}, M. R\^{u}\v{z}i\v{c}ka, Lebesgue and Sobolev Spaces with Variable Exponents. in: Lecture Notes
in Mathematics, vol. 2017. Springer-Verlag. Heidelberg, 2011.
\bibitem{29} M. R\^{u}\v{z}i\v{c}ka, Electrorheological Fluids: Modeling and Mathematical Theory. in: Lectures Notes in Mathematics, vol. 1748,
Springer-Verlag, Berlin, 2000.
\bibitem{01} R. Aboulaich, D. Meskine, A. Souissi, New diffusion models in image processing. Computers and Mathematics with Applications. \textbf{56}%
(2008), 874-882.
\bibitem{21} Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising. Nonlinear Analysis: Real World
Applications.\textbf{12}(2011), 2904-2918.
\bibitem{35} L. Songzhe, G. Wenjie, C. Chunling and Y. Hongjun, Study of the solutions to a model porous medium equation with variable exponent of
nonlinearity. J. Math. Anal. Appl. \textbf{342}(2008), 27-38.
\bibitem{7} P. Bernner, W. Von Whal, Global classical solutions of nonlinear wave equations, Math. Z. 176 (1981) 87--121.
\bibitem{20} L.E. Payne, D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975) 273--303.
\bibitem{210} H. Pecher, Die existenz regul\"{a}er L\"{o}sungen f\"{u}r Cauchy-und anfangs-randwertproble-me michtlinear wellengleichungen, Math. Z.
140 (1974) 263--279.
\bibitem{220} D.H. Sattinger, On global solutions for nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30 (1968) 148--172.
\bibitem{26} B.X. Wang,\ Nonlinear scattering theory for a class of wave equations in $H^{s}$, J. Math. Anal. Appl. 296 (2004) 74--96.
\bibitem{16} C.X. Miao, The time space estimates and scattering at low energy for nonlinear higher order wave equations, Acta Math. Sin. Ser. A 38
(1995) 708--717.
\bibitem{18} M. Nakao, Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term, J. Math.
Soc. Japan 30 (1978) 375--394.
\bibitem{19} M. Nakao, H. Kuwahara, Decay estimates for some semilinear wave equations with degenerate dissipative terms, Funkcial. Ekvac. 30 (1987)
135--145.
\bibitem{28} Y.J. Ye, Existence and asymptotic behavior of global solutions for a class of nonlinear higher-order wave equation, J. Inequal. Appl. 2010
(2010) 1--14.
\bibitem{100} A.B. Aliev, B.H. Lichaei, Existence and non-existence of global solutions of the Cauchy problem for higher order semilinear
pseudo-hyperbolic equations, Nonlinear Anal. TMA 72 (2010) 3275--3288.
\bibitem{Acerbi4} Acerbi E, Mingione G. Regularity results for electrorheological fluids, the stationary case, C. R. Acad. Sci. Paris,
\textbf{334}(2002), 817--822.
\bibitem{Diening1} Diening L, R\r{u}\v{z}i\v{c}ka M. Calderon Zygmund operators on generalized Lebesgue spaces\emph{\ }$\emph{L}^{p(x)}(\Omega )$
and problems related to fluid dynamics. Preprint Mathematische Fakult\"{a}t, Albert-Ludwigs-Universit\"{a}t Freiburg, \textbf{120 }(2002), 197--220.
\bibitem{Halsey} Halsey T.C. Electrorheological fluids, Science, \textbf{258} (1992), 761--766.
\bibitem{Levin} H.A. Levin, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form\emph{\ Pu}$_{\emph{t}}$%
\emph{\ = -Au+F (u)}, Arch. Ration. Mech. Anal. 51 (1973) 371--386.
\bibitem{Ali} Ali, K, Khadijeh, B. Blow-up in a semilinear parabolic problem with variable source under positive initial energy\emph{, }Applicable
Analysis, 2014.
\bibitem{Lions1} Lions, J.L. Quelques m\'{e}thodes de r\'{e}solution des probl\`{e}mes aux limites non lin\'{e}aires. Dunod, Paris, 1966.
\bibitem{9} Al-Mahdi, A.M.; Al-Gharabli, M.M.; Noor, M.; Audu, J.D. Stability Results for a Weakly Dissipative Viscoelastic Equation with
Variable-Exponent Nonlinearity: Theory and Numerics. Math. Comput. Appl.
2023, 28, 5.
\bibitem{10} Park, S.H.; Kang, J.R. Blow-up of solutions for a viscoelastic wave equation with variable exponents. Math. Meth. Appl. Sci. 2019, 42,
2083--2097.
\bibitem{11} Piskin, E. Blow up of solutions for a nonlinear viscoelastic wave equations with variable exponents. Middle East J. Sci. 2019, 5,
134--145.
\bibitem{park} Park S-H. General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities. Bound Value Probl.
2022;2022:23.
Published
2026-04-02
Section
Research Articles
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