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On p-Kirchhoff-type parabolic problems

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Abstract

In this work, the existence result to a p-Kirchhoff-type parabolic system is considered. Based on Galerkin’s method and the theory of Young measures, we prove the existence of weak solutions.

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References

  1. Adams, R.: Sobolev Spaces. Ac Press, New York (1975)

    MATH  Google Scholar 

  2. Azroul, E., Balaadich, F.: Weak solutions for generalized p-Laplacian systems via Young measures. Moroccan J. Pure Appl. Anal. (MJPAA) 4(2), 77–84 (2018)

    Article  Google Scholar 

  3. Azroul, E., Balaadich, F.: Quasilinear elliptic systems in perturbed form. Int. J. Nonlinear Anal. Appl. 10(2), 255–266 (2019)

    MATH  Google Scholar 

  4. Azroul, E., Balaadich, F.: A weak solution to quasilinear elliptic problems with perturbed gradient. Rend. Circ. Mat. Palermo (2) (2020). https://doi.org/10.1007/s12215-020-00488-4

    Article  MATH  Google Scholar 

  5. Azroul, E., Balaadich, F.: On strongly quasilinear elliptic systems with weak monotonicity. J. Appl. Anal. (2021). https://doi.org/10.1515/jaa-2020-2041

    Article  MathSciNet  MATH  Google Scholar 

  6. Azroul, E., Balaadich, F.: Strongly quasilinear parabolic systems in divergence form with weak monotonicity. Khayyam J. Math. 6(1), 57–72 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Azroul, E., Balaadich, F.: Existence of solutions for a class of Kirchhoff-type equation via Young measures. Numer. Funct. Anal. Optim. (2021). https://doi.org/10.1080/01630563.2021.1885044

    Article  MathSciNet  MATH  Google Scholar 

  8. Balaadich, F., Azroul, E.: Elliptic systems of \(p\)-Laplacian type. Tamkang J. Math. (2021). https://doi.org/10.5556/j.tkjm.53.2022.3296

    Article  MATH  Google Scholar 

  9. Ball, J.M.: A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions (Nice, 1988). Lecture Notes in Physics, vol. 344, pp. 207–215 (1989)

  10. Chen, C., Kuo, Y., Wu, T.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250(4), 1876–1908 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, S., Zhang, B., Tang, X.: Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity. Adv. Nonlinear Anal. 9(1), 148–167 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chipot, M., Valente, V., Caffarelli, G.V.: Remarks on a nonlocal problems involving the Dirichlet energy. Rend. Sem. Math. Univ. Padova 110, 199–220 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Chipot, M., Savitska, T.: Nonlocal p-Laplace equations depending on the lp norm of the gradient. Adv. Differ. Equ. 19, 997–1020 (2014)

    MATH  Google Scholar 

  14. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Tata McGraw-Hill Education, London (1955)

    MATH  Google Scholar 

  15. Corrêa, F.J.S.A., Figueiredo, G.M.: On a elliptic equation of p-Kirchhoff type via variational methods. Bull. Austral. Math. Soc. 74, 263–277 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Corrêa, F.J.S.A., Figueiredo, G.M.: On a p-Kirchhoff equation via Krasnoselskii’s genus. Appl. Math. Lett. 22, 819–822 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dolzmann, G., Hungerühler, N., Muller, S.: Nonlinear elliptic systems with measure-valued right hand side. Math. Z. 226, 545–574 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations, vol. 74. American Mathematical Society, Providence (1990)

    Google Scholar 

  19. Ghisi, M., Gobbino, M.: Hyperbolic-parabolic singular perturbation for middly degenerate Kirchhoff equations: time-decay estimates. J. Differ. Equ. 245, 2979–3007 (2008)

    Article  MATH  Google Scholar 

  20. Han, Y.Z., Li, Q.W.: Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Comput. Math. Appl. 75, 3283–3297 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

    MATH  Google Scholar 

  22. Landes, R.: On the existence of weak solutions for quasilinear parabolic boundary problems. Proc. R. Soc. Edinb. Sect. A 89, 217–237 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, J., Han, Y.: Global existence and finite time blow-up of solutions to a nonlocal p-Laplace equation. Math. Model. Anal. 24(2), 195–217 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, H.: Blow-up of solutions to a p-Kirchhoff-type parabolic equation with general nonlinearity. J. Dyn. Control Syst. (2019). https://doi.org/10.1007/s10883-019-09463-4

    Article  Google Scholar 

  25. Lieberman, G.M.: The natural generalizationj of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations. Commun. Partial Differ. Equ. 16(2 & 3), 311–361 (1991)

    Article  MATH  Google Scholar 

  26. Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proceedings of International Symposium, Institute of Mathematics (IM) at the Universidade Federal Rio de Janeiro, Rio de Janeiro, 1977), pp. 284–346. North-Holland Mathematical Studies, North-Holland (1978)

  27. Liu, D., Zhao, P.: Multiple nontrivial solutions to a p-Kirchhoff equation. Nonlinear Anal. 75, 5032–5038 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mingqi, X., Rǎdulescu, V.D., Zhang, B.: Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities. ESAIM Control Optim. Calc. Var. 24(3), 1249–1273 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mingqi, X., Rǎdulescu, V.D., Zhang, B.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58(2), 57 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pucci, P., Xiang, M., Zhang, B.: A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete Contin. Dyn. Syst. 37(7), 4035–4051 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 30(2), 148–172 (1968)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Balaadich, F. On p-Kirchhoff-type parabolic problems. Rend. Circ. Mat. Palermo, II. Ser 72, 1005–1016 (2023). https://doi.org/10.1007/s12215-021-00705-8

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