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.
Similar content being viewed by others
References
Adams, R.: Sobolev Spaces. Ac Press, New York (1975)
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)
Azroul, E., Balaadich, F.: Quasilinear elliptic systems in perturbed form. Int. J. Nonlinear Anal. Appl. 10(2), 255–266 (2019)
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
Azroul, E., Balaadich, F.: On strongly quasilinear elliptic systems with weak monotonicity. J. Appl. Anal. (2021). https://doi.org/10.1515/jaa-2020-2041
Azroul, E., Balaadich, F.: Strongly quasilinear parabolic systems in divergence form with weak monotonicity. Khayyam J. Math. 6(1), 57–72 (2020)
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
Balaadich, F., Azroul, E.: Elliptic systems of \(p\)-Laplacian type. Tamkang J. Math. (2021). https://doi.org/10.5556/j.tkjm.53.2022.3296
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)
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)
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)
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)
Chipot, M., Savitska, T.: Nonlocal p-Laplace equations depending on the lp norm of the gradient. Adv. Differ. Equ. 19, 997–1020 (2014)
Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Tata McGraw-Hill Education, London (1955)
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)
Corrêa, F.J.S.A., Figueiredo, G.M.: On a p-Kirchhoff equation via Krasnoselskii’s genus. Appl. Math. Lett. 22, 819–822 (2009)
Dolzmann, G., Hungerühler, N., Muller, S.: Nonlinear elliptic systems with measure-valued right hand side. Math. Z. 226, 545–574 (1997)
Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations, vol. 74. American Mathematical Society, Providence (1990)
Ghisi, M., Gobbino, M.: Hyperbolic-parabolic singular perturbation for middly degenerate Kirchhoff equations: time-decay estimates. J. Differ. Equ. 245, 2979–3007 (2008)
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)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Landes, R.: On the existence of weak solutions for quasilinear parabolic boundary problems. Proc. R. Soc. Edinb. Sect. A 89, 217–237 (1981)
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)
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
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)
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)
Liu, D., Zhao, P.: Multiple nontrivial solutions to a p-Kirchhoff equation. Nonlinear Anal. 75, 5032–5038 (2012)
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)
Mingqi, X., Rǎdulescu, V.D., Zhang, B.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58(2), 57 (2019)
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)
Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 30(2), 148–172 (1968)
Zheng, S., Chipot, M.: Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asymptot. Anal. 45, 301–312 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-021-00705-8