Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2021, Cilt: 50 Sayı: 2, 397 - 413, 11.04.2021
https://doi.org/10.15672/hujms.657295

Öz

Kaynakça

  • [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] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Q. J. Math. Oxf. Ser. 28, 473–486, 1977.
  • [3] S.S. Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs: Victoria Univ, 2002.
  • [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] 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] 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] 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] P. Martinez, A new method to obtain decay rate estimates for dissipative system, ESAIM Control OPTİM. Calc. Var. 4, 419–444, 1999.
  • [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] 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] 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] 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] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149, 155–182, 1999.

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

Yıl 2021, Cilt: 50 Sayı: 2, 397 - 413, 11.04.2021
https://doi.org/10.15672/hujms.657295

Öz

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].

Kaynakça

  • [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] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Q. J. Math. Oxf. Ser. 28, 473–486, 1977.
  • [3] S.S. Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs: Victoria Univ, 2002.
  • [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] 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] 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] 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] P. Martinez, A new method to obtain decay rate estimates for dissipative system, ESAIM Control OPTİM. Calc. Var. 4, 419–444, 1999.
  • [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] 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] 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] 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] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149, 155–182, 1999.
Toplam 13 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Erhan Pişkin 0000-0001-6587-4479

Fatma Ekinci 0000-0002-9409-3054

Yayımlanma Tarihi 11 Nisan 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 50 Sayı: 2

Kaynak Göster

APA Pişkin, E., & Ekinci, F. (2021). Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities. Hacettepe Journal of Mathematics and Statistics, 50(2), 397-413. https://doi.org/10.15672/hujms.657295
AMA Pişkin E, Ekinci F. Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities. Hacettepe Journal of Mathematics and Statistics. Nisan 2021;50(2):397-413. doi:10.15672/hujms.657295
Chicago Pişkin, Erhan, ve Fatma Ekinci. “Qualitative Analysis of Solutions for a Kirchhoff-Type Parabolic Equation With Multiple Nonlinearities”. Hacettepe Journal of Mathematics and Statistics 50, sy. 2 (Nisan 2021): 397-413. https://doi.org/10.15672/hujms.657295.
EndNote Pişkin E, Ekinci F (01 Nisan 2021) Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities. Hacettepe Journal of Mathematics and Statistics 50 2 397–413.
IEEE E. Pişkin ve F. Ekinci, “Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities”, Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 2, ss. 397–413, 2021, doi: 10.15672/hujms.657295.
ISNAD Pişkin, Erhan - Ekinci, Fatma. “Qualitative Analysis of Solutions for a Kirchhoff-Type Parabolic Equation With Multiple Nonlinearities”. Hacettepe Journal of Mathematics and Statistics 50/2 (Nisan 2021), 397-413. https://doi.org/10.15672/hujms.657295.
JAMA Pişkin E, Ekinci F. Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities. Hacettepe Journal of Mathematics and Statistics. 2021;50:397–413.
MLA Pişkin, Erhan ve Fatma Ekinci. “Qualitative Analysis of Solutions for a Kirchhoff-Type Parabolic Equation With Multiple Nonlinearities”. Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 2, 2021, ss. 397-13, doi:10.15672/hujms.657295.
Vancouver Pişkin E, Ekinci F. Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities. Hacettepe Journal of Mathematics and Statistics. 2021;50(2):397-413.