Global existence and blow-up of solutions for a $$\mathtt {p}$$ -Kirchhoff type parabolic equation with logarithmic nonlinearity
Tóm tắt
In this paper a class of
$$\mathtt {p}$$
-Kirchhoff type parabolic equation with logarithmic nonlinearity is considered. By applying Galerkin’s approximation and the modified potential well method, some sufficient conditions are obtained for the existence of global and finite blow up of solutions.
Tài liệu tham khảo
Chipot, M., Savitska, T.: Nonlocal \({p}\)-Laplace equations depending on the \({L}^{p}\) norm of the gradient. Adv. Differ. Equ. 19(11–12), 997–1020 (2014)
D’Ancona, P., Shibata, Y.: on global solvability of non-linear viscoelastic equation in the analytic category. Math. Methods Appl. Sci. 17, 477–489 (1994)
D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)
Fu, Y.Q., Xiang, M.Q.: Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent. Appl. Anal. 95(3), 524–544 (2016). https://doi.org/10.1080/00036811.2015.1022153
Ghisi, M., Gobbino, M.: Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: time-decay estimates. J. Differ. Equ. 245(10), 2979–3007 (2008). https://doi.org/10.1016/j.jde.2008.04.017
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(9), 3283–3297 (2018). https://doi.org/10.1016/j.camwa.2018.01.047
Hou, A.J., Liao, J.F.: Existence of positive solution for a class of nonlocal problem with strong singularity and linear term. Differ. Equ. Appl. 12(3), 277–290 (2020)
Khuddush, M., Prasad, K.R., Botta, B.: Local existence and blow up of solutions for a system of viscoelastic wave equations of Kirchhoff type with delay and logarithmic nonlinearity. Int. J. Math. Model. Comput. 11(3), 1–11 (2021)
Khuddush, M., Prasad, K.R.: Positive solutions for an iterative system of nonlinear elliptic equations. Bull. Malays. Math. Sci. Soc. 45, 245–272 (2022). https://doi.org/10.1007/s40840-021-01183-y
Khuddush, M., Prasad, K.R., Bharathi, B.: Global existence and blowup of solutions for a semilinear Klein–Gordon equation with the product of logarithmic and power-type nonlinearity. Ann. Univ. Ferrara 68, 187–201 (2022). https://doi.org/10.1007/s11565-022-00395-9
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Levine, H.A.: Some nonexistence and instability theorems for solutions of formally parabolic equation of the form \(pu_t=-au+Fu\). Arch. Ration. Mech. Anal. 51(5), 371–386 (1973). https://doi.org/10.1007/BF00263041
Li, J., Han, Y.: Global existence and finite time blow-up of solutions to a nonlocal pLaplace equation. Math. Model. Anal. 24(2), 195–217 (2019). https://doi.org/10.3846/mma.2019.014
Liu, Y.C.: On potential wells and vacuum isolating of solutions for semilinear wave equations. J. Differ. Equ. 192(1), 155–169 (2003)
Liu, Y.C., Zhao, J.S.: On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. 64, 2665–2687 (2006)
Nishihara, K.: On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 7, 437–459 (1984)
Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 30(2), 148–172 (1968). https://doi.org/10.1007/BF00250942
Sun, Y.J., Tan, Y.X.: Kirchhoff type equations with strong singularities. Commun. Pure Appl. Anal. 18(1), 181–193 (2019)