Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities

Abstract

In this work, the local and global existence of weak solutions by using the Faedo-Galerkin method, the finite time blow up of the weak solution with positive initial energy and the general decay of the solution are discussed. Finally, we consider the exponential growth of the solution with sufficient conditions. This work generalizes and improves earlier results in the literature, see [L.X. Truong and N. Van Y, On a class of nonlinear heat equations with viscoelastic term, Comput. Math. Appl., 2016] and [L.X. Truong and N. Van Y, Exponential growth with ${L}^{p}$-norm of solutions for nonlinear heat equations with viscoelastic term, Appl. Math. Comput., 2016].

Keywords

• Existence of solutions
• Blow up
• Exponential growth
• Decay
• Kirchhoff-type equation

References

1. [1] L. Alfonsi and F. Weissler, Blow-up in $R^{n}$ for a parabolic equation with a damping nonlinear gradient term, Progr. Nonlinear Differential Equations Appl. 7, 1–20, 1992.
2. [2] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Q. J. Math. Oxf. Ser. 28, 473–486, 1977.
3. [3] S.S. Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs: Victoria Univ, 2002.
4. [4] Y. He, H. Gao and H. Wang, Blow-up and decay for a class of pseudo-parabolic p- Laplacian equation with logarithmic nonlinearity, Comput. Math. Appl. 75, 459–469, 2018.
5. [5] V.K. Kalantarov and O. A. Ladyzhenskaya, The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Sov. Math. 10, 53–70, 1978.
6. [6] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, Vol. 28, Amer. Math. Soc., 1968.
7. [7] H. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t}=-Au+\mathcal{F} (u)$, Arch. Ration. Mech. Anal. 51, 371–386, 1973.
8. [8] P. Martinez, A new method to obtain decay rate estimates for dissipative system, ESAIM Control OPTİM. Calc. Var. 4, 419–444, 1999.

9. [9] S.A. Messaoudi, Blow-up of semilinear heat equation with a visco-elastic term, Progr. Nonlinear Differential Equations Appl. 64, 351–356, 2005.
10. [10] N. Polat, Blow up of solution for a nonlinear reaction diffusion equation with multiple nonlinearities, Int. J. Sci. Technol. 2 (2), 123–128, 2007.
11. [11] L.X. Truong and N. Van Y, On a class of nonlinear heat equations with viscoelastic term, Comput. Math. Appl. 72, 216–232, 2016.
12. [12] L.X. Truong and N. Van Y, Exponential growth with $L^{p}$-norm of solutions for nonlinear heat equations with viscoelastic term, Appl. Math. Comput. 273, 656–663, 2016.
13. [13] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149, 155–182, 1999.