Well-posedness results for a class of nonlinear reaction-diffusion equations with memory




nonlinear heat equation, reaction-diffusion equations, memory term, blow-up


In this paper, we are interested to study the initial value problem for nonlinear heat equation with memory.  For our problem, we show that the local existence theory related to the finite time blow-up is also obtained for the problem with logarithmic nonlinearity. We also obtain the blowup continuation property of the mild solution. The principal techniques based on some Sobolev embeddings with \(L^p\) spaces.  


C.M. Chen, V. Thomée, The lumped mass finite element method for a parabolic problem, J. Austral. Math. Soc. Ser. B, 26(3) (1985), 329-354. DOI: https://doi.org/10.1017/S0334270000004549

V. Shelukhin, A non-local in time model for radionuclides propagation inStokes fluids Dynamics of Fluids with Free Boundaries, Siberian Branch of the Russian Academy of Sciences Institute of Hydrodynamics, 107, pp. 180-193.

H. Engler, Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Math. Z., 202(2) (1989), pp. 251-259. DOI: https://doi.org/10.1007/BF01215257

M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31 (1968), pp. 113-126. DOI: https://doi.org/10.1007/BF00281373

J. Kemppainen, J. Siljander, V. Vergara, R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $R^d$, Math. Ann., 366 (2016), pp. 941-979. DOI: https://doi.org/10.1007/s00208-015-1356-z

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal. 21(5) (1990), pp. 1213-1224. DOI: https://doi.org/10.1137/0521066

J.M. Rivera, L.H. Fatori, Smoothing effect and propagations of singularities for viscoelastic plates, J. Math. Anal. App., 206 (1997), pp. 397-427. DOI: https://doi.org/10.1006/jmaa.1997.5223

F.B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), pp. 29-40. DOI: https://doi.org/10.1007/BF02761845

V. Keyantuo, C. Lizama, Holder Continuous solutions for integro-differential equations and maximal regularity, J. Differential Equations 230 (2006), pp. 634-660. DOI: https://doi.org/10.1016/j.jde.2006.07.018

M. Conti, S. Gatti, M. Grasselli, V. Pata, Two-dimensional reaction-diffusion equations with memory, Quart. Appl. Math., 68(4) (2010), pp. 607-643. DOI: https://doi.org/10.1090/S0033-569X-2010-01167-7

S. Gatti, A. Miranville, V. Pata, S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140(2) (2010), pp. 329-366. DOI: https://doi.org/10.1017/S0308210509000365

V. Barbu, S. Sritharan, Navier-Stokes equation with hereditary viscosity, Z. Angew. Math. Phys., 54 (2003), pp. 449-461. DOI: https://doi.org/10.1007/s00033-003-1087-y

T. Caraballo, J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), pp. 271-297. DOI: https://doi.org/10.1016/j.jde.2004.04.012

I. Munteanu, Boundary stabilization of the Navier-Stokes equation with fading memory, Int. J. Control 88(3) (2015), pp. 531-542. DOI: https://doi.org/10.1080/00207179.2014.964780

M. D'Abbico, The influence of a nonlinear memory on the damped wave equation, Nonlinear Anal., 95 (2014), pp. 130-145. DOI: https://doi.org/10.1016/j.na.2013.09.006

M. Fabrizio, S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), pp. 1245-1264. DOI: https://doi.org/10.1080/0003681021000035588

A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl. 382 (2011), pp. 748-760. DOI: https://doi.org/10.1016/j.jmaa.2011.04.079

M.L. Heard, S. M.Rankin III, A semilinear parabolic Volterra integro-differential equation, J. Differential Equations, 71(2) (1988), pp. 201-233. DOI: https://doi.org/10.1016/0022-0396(88)90023-X

A. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in a unbounded domain, J. Differential Equations, 225 (2006), pp. 528-548. DOI: https://doi.org/10.1016/j.jde.2005.12.001

I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems and application to wave and plate boundary control problems, Appl. Math. Optim. 23 (1991), pp. 109-154. DOI: https://doi.org/10.1007/BF01442394

M. A. Jorge Silva and T. F. Ma, On a viscoelastic plate equation with history setting and pertubation of p-Laplacian type, IMA J. Appl. Math., 78 (2013), pp. 1130-1146. DOI: https://doi.org/10.1093/imamat/hxs011

B. de Andrade, A. Viana, Abstract Volterra integro-differential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), pp. 1131-1175. DOI: https://doi.org/10.1007/s00208-016-1469-z

T. Cazenave, F. Dickstein, F.B. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal., 68(4) (2008), pp. 862-874. DOI: https://doi.org/10.1016/j.na.2006.11.042

B. De Andrade, A. Viana, On a fractional reaction-diffusion equation, Z. Angew. Math. Phys., 68 (2017), 11 pages. DOI: https://doi.org/10.1007/s00033-017-0801-0

J.M. Arrieta, A.N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352(1) (1999), pp. 285-310. DOI: https://doi.org/10.1090/S0002-9947-99-02528-3

B. de Andrade, A. Viana, Integro-differential equations with applications to a plate equation with memory, Math. Nachr., (2016), pp. 1-14.

J.M. Arrieta, A.N. Carvalho, A. Rodriguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations, 156 (1999), pp. 376-406. DOI: https://doi.org/10.1006/jdeq.1998.3612

H. Brezis, T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996) pp. 277-304. DOI: https://doi.org/10.1007/BF02790212

Y. Giga, A bound for global solutions of semilinear heat equations, Commun. Math. Phys., 103(3) (1986), pp. 415-421. DOI: https://doi.org/10.1007/BF01211756

M.G. Grillakis, Regularity and asymptotic behavior of the wave equation with critical nonlinearity, Ann. Math., 132 (1990), pp. 485-509. DOI: https://doi.org/10.2307/1971427

H. Chen, P. Luo, G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422(1) (2015), pp. 84-98. DOI: https://doi.org/10.1016/j.jmaa.2014.08.030

J. Barrow, P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D, 52 (1995), pp. 5576-5587. DOI: https://doi.org/10.1103/PhysRevD.52.5576

Y. Chen, R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192(2020), 111664, 39 pages. DOI: https://doi.org/10.1016/j.na.2019.111664

N.S. Papageorgiouae, V.D. Radulescu, D.D. Repova, Positive solutions for nonlinear Neumann problems with singular terms and convection, Journal de Mathematiques Pures et Appliquees, 136 (2020), pp. 1-21. DOI: https://doi.org/10.1016/j.matpur.2020.02.004

X. Wang, Y. Chen, Y. Yang, J.Li, R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019) (2019), pp. 475-499. DOI: https://doi.org/10.1016/j.na.2019.06.019

L.C. Evans, Partial Differential Equations, 19 (1998), (Providence, RI: American Mathematical Society).

G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55(2) (2017), pp. 472-495. DOI: https://doi.org/10.1137/15M1033952

E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136(5) (2012), pp. 521-573. DOI: https://doi.org/10.1016/j.bulsci.2011.12.004

M. Kafini. S. Messaoudi, Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay, Appl. Anal., 99(3) (2020), pp. 530-547. DOI: https://doi.org/10.1080/00036811.2018.1504029

E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, second printing corrected, Springer-Verlag, Berlin, 1969.

I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, Methods of Their Solution and Some of Their Applications, 198 (1998), Elsevier, Amsterdam.



2023 May 07

How to Cite

D. B. Ho, D. P. Nguyen, and V. T. Vo, “Well-posedness results for a class of nonlinear reaction-diffusion equations with memory”, Electron. J. Appl. Math., vol. 1, no. 1, pp. 1–29, May 2023.



Research Article
Received 2023 May 25
Accepted 2023 May 04
Published 2023 May 07

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