Skip to main content
SpringerLink
Log in

On the Well-Posedness and Blow-Up for a Semilinear Biparabolic Equation

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the Cauchy problem for a semilinear biparabolic equation

$$\begin{aligned} \left( \partial _t + {\mathcal {A}} \right) ^2 u = G(x,t;u), \quad x \in \Omega ,~ t\ge 0. \end{aligned}$$

Results of the local well-posedness (local existence, regularity, and continuous dependence) are given when G is globally Lipschitz. Also, the existence for large times (continuation) of the solutions and a finite time blow-up results are proposed when G is locally Lipschitz functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alvarez, E., Gal, C.G., Keyantuo, V., Warma, M.: Well-posedness results for a class of semi-linear super-diffusive equations. Nonlinear Anal. 181, 24–61 (2019)

    Article  MathSciNet  Google Scholar 

  2. Andrade, B., Au, V.V., O’Regan, D., Tuan, N.H.: Well-posedness results for a class of semilinear time fractional diffusion equations. Z. Angew. Math. Phys. 71(161), 23 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Au, V.V., Kim, H.T.V., Anh, N.T.: On a class of semilinear nonclassical fractional wave equations with logarithmic nonlinearity. Math. Meth. Appl. Sci. 44(14), 11022–11045 (2021)

    Article  MathSciNet  Google Scholar 

  4. Au, V.V., Singh, J., Anh, N.T.: Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electron. Res. Arch. 29(6), 3581–3607 (2021)

    Article  MathSciNet  Google Scholar 

  5. Besma, K., Nadji, B., Faouzia, R.: A modified quasi-boundary value method for an abstract ill-posed biparabolic problem. Open Math. 15, 1649–1666 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bulavatsky, V.M.: Mathematical modeling of filtrational consolidation of soil under motion of saline solutions on the basis of biparabolic model. J. Autom. Inform. Sci. 35(8), 13–22 (2003)

    Google Scholar 

  7. Bulavatsky, V.M.: Fractional differential analog of biparabolic evolution equation and some its applications. Cybern. Syst. Anal. 52(5), 337–347 (2016)

    Article  Google Scholar 

  8. Bulavatsky, V.M., Skopetsky, V.V.: Generalized mathematical model of the dynamics of consolidation processes with relaxation. Cybern. Syst. Anal. 44(5), 646–654 (2008)

    Article  MathSciNet  Google Scholar 

  9. Can, N.H., Tuan, N.H., O’Regan, D., Au, V.V.: On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evol. Equ. Control Theory 10(1), 103–127 (2021)

    Article  MathSciNet  Google Scholar 

  10. Carvalho, A.N., Gentile, C.B.: Asymptotic behaviour of non-linear parabolic equations with monotone principal part. J. Math. Anal. Appl. 280(2), 252–272 (2003)

    Article  MathSciNet  Google Scholar 

  11. Chafee, N.: Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. J. Differ. Equ. 18(1), 111–134 (1975)

    Article  MathSciNet  Google Scholar 

  12. Dafermos, C.M., Slemrod, M.: Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13(1), 97–106 (1973)

    Article  MathSciNet  Google Scholar 

  13. Fushchich, V.L., Galitsyn, A.S., Polubinskii, A.S.: A new mathematical model of heat conduction processes. Ukrainian Math. J. 42, 210–216 (1990)

    Article  MathSciNet  Google Scholar 

  14. Galaktionov, V.A., Vazquez, J.L.: Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach. J. Funct. Anal. 100(2), 435–462 (1991)

    Article  MathSciNet  Google Scholar 

  15. Gatti, S., Grasselli, M., Pata, V., Miranville, A.: Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D. Math. Models Methods Appl. Sci. 15(2), 165–198 (2005)

    Article  MathSciNet  Google Scholar 

  16. Henry, D.: Geometric theory of semilinear parabolic equations. Lect. Notes Math., vol. 840. Springer-Verlag, Berlin (1981)

    Book  Google Scholar 

  17. Joseph, L., Preziosi, D.D.: Heat waves. Rev. Mod. Phys. 41–73 (1989)

  18. Kalantarov, V., Zelik, S.: Finite-dimensional attractors for the quasi-linear strongly-damped wave equation. J. Differ. Equ. 247(4), 1120–1155 (2009)

    Article  MathSciNet  Google Scholar 

  19. Lakhdari, A., Boussetila, N.: An iterative regularization method for an abstract ill-posed biparabolic problem. Bound. Value Probl. 55, 1–17 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Marras, M., Piro, S.V.: Bounds for blow-up time in nonlinear parabolic systems. Disctete Contin. Dyn. Syst. Ser. A 1025–1031 (2011)

  21. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  22. Mottoni, P., Tesei, A., Pucci, C., Payne, L.E.: Asymptotic stability results for system of quasilinear parabolic equations. Appl. Anal. 9, 7–21 (1979)

    Article  MathSciNet  Google Scholar 

  23. Pata, V., Squassina, M.: On the strongly damped wave equation. Commun. Math. Phys. 253(3), 511–533 (2005)

    Article  MathSciNet  Google Scholar 

  24. Payne, L.E.: On a proposed model for heat conduction. IMA J. Appl. Math. 71, 590–599 (2006)

    Article  MathSciNet  Google Scholar 

  25. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  26. Trong, D.D., Nane, E., Minh, N.D., Tuan, N.H.: Continuity of solutions of a class of fractional equations. Potential Anal. 49, 423–478 (2018)

    Article  MathSciNet  Google Scholar 

  27. Tuan, N.H., Au, V.V., Tri, V.V., O’Regan, D.: On the well-posedness of a nonlinear pseudo-parabolic equation. J. Fixed Point Theory Appl. 22(77) (2020), 21 pages

  28. Tuan, N.H., Kirane, M., Nam, D.H.Q., Au, V.V.: Approximation of an inverse initial problem for a biparabolic equation. Mediterr. J. Math. 15(18) (2018), 18 pages

  29. Tuan, N.H., Au, V.V., Xu, R., Wang, R.: On the initial and terminal value problem for a class of semilinear strongly material damped plate equations. J. Math. Anal. Appl. 492(2), 124481 (2020)

    Article  MathSciNet  Google Scholar 

  30. Wang, L., Zhou, X., Wei, X.: Heat Conduction: Mathematical Models and Analytical Solutions. Springer, Berlin (2008)

    MATH  Google Scholar 

  31. Webb, J.R.L.: Weakly singular Gronwall inequalities and applications to fractional differential equations. J. Math. Anal. Appl. 471(1–2), 692–711 (2019)

    Article  MathSciNet  Google Scholar 

  32. Winkler, M.: Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening. J. Differ. Equ. 257(4), 1056–1077 (2014)

    Article  MathSciNet  Google Scholar 

  33. Yang, D., Wang, J.R., O’Regan, D.: Asymptotic properties of the solutions of nonlinear non-instantaneous impulsive differential equations. J. Franklin Inst. 354(15), 6978–7011 (2017)

    Article  MathSciNet  Google Scholar 

  34. Yang, Y., Salik Ahmed, Md., Qin, L., Xu, R.: Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations. Opuscula Math. 39(2), 297–313 (2019)

    Article  MathSciNet  Google Scholar 

  35. Zhang, B.G., Zhou, Y., Huang, Y.Q.: Existence of positive solutions for certain nonlinear partial difference equations. Math. Comput. Model. 38(3–4), 331–337 (2003)

    Article  MathSciNet  Google Scholar 

  36. Zheng, S., Chipot, M.: Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asymptot. Anal. 45(3–4), 301–312 (2005)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous reviewers for his/her careful reading of our manuscript and his/her insightful comments and suggestions. This work is funded by Van Lang University.

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

    Vo Van Au

  2. Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    Vo Van Au

  3. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China

    Yong Zhou

  4. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Yong Zhou

  5. School of Mathematical and Statistical Sciences, National University of Ireland, Galway, Ireland

    Donal O’Regan

Authors
  1. Vo Van Au
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vo Van Au.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Au, V., Zhou, Y. & O’Regan, D. On the Well-Posedness and Blow-Up for a Semilinear Biparabolic Equation. Mediterr. J. Math. 19, 35 (2022). https://doi.org/10.1007/s00009-021-01970-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-021-01970-8

Keywords

Mathematics Subject Classification

Search

Navigation