Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

  • Received: 01 January 2021 Revised: 01 July 2021 Published: 07 September 2021
  • Primary: 35K70, 35B40; Secondary: 35K35

  • In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy $ J(u_0)\leq d $. When the initial energy $ J(u_0)>d $, we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.

    Citation: Yang Cao, Qiuting Zhao. Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations[J]. Electronic Research Archive, 2021, 29(6): 3833-3851. doi: 10.3934/era.2021064

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  • In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy $ J(u_0)\leq d $. When the initial energy $ J(u_0)>d $, we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.



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    [1] Global nonexistence for nonlinear Kirchhoff systems. Arch. Ration. Mech. Anal. (2010) 196: 489-516.
    [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
    [3] Blow-up in a partial differential equation with conserved first integral. SIAM J. Appl. Math. (1993) 53: 718-742.
    [4] Y. Cao and Q. Zhao, Asymptotic behavior of global solutions to a class of mixed pseudo-parabolic Kirchhoff equations, Appl. Math. Lett., 118 (2021), 107119, 6 pp. doi: 10.1016/j.aml.2021.107119
    [5] Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness. Nonlinear Dynam. (2016) 84: 35-50.
    [6] Digital removal of random media image degradations by solving the diffusion equation backwards in time. SIAM J. Numer. Anal. (1978) 15: 344-367.
    [7] Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete Contin. Dyn. Syst. (2019) 39: 1185-1203.
    [8] Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity. Adv. Nonlinear Anal. (2020) 9: 148-176.
    [9] Variable exponent linear growth functionals in image restoration. SIMA. J. Appl. Math. (2006) 66: 1383-1406.
    [10] Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity. J. Math. Anal. Appl. (2019) 478: 393-420.
    [11] Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent. Appl. Anal. (2016) 95: 524-544.
    [12] Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy. Comput. Math. with Appl. (2018) 76: 2477-2483.
    [13] Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Comput. Math. Appl. (2018) 75: 3283-3297.
    [14] Existence, multiplicity and nonexistence results for Kirchhoff type equations. Adv. Nonlinear Anal. (2021) 10: 616-635.
    [15] Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire (2008) 25: 215-218.
    [16] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
    [17] Global existence and finite time blow-up of solutions to a nonlocal $p$-Laplace equation. Math. Model. Anal. (2019) 24: 195-217.
    [18] Global existence and blow up of solutions for pseudo-parabolic equation with singular potential. J. Differential Equations (2020) 269: 4914-4959.
    [19] On potential wells and vacuum isolating of solutions for semilinear wave equations. J. Differential Equations (2003) 192: 155-169.
    [20] On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. (2006) 64: 2665-2687.
    [21] Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations. Commun. in Partial Differential Equations (1998) 23: 457-486.
    [22] Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. (1975) 22: 273-303.
    [23] C. Qu and W. Zhou, Asymptotic analysis for a pseudo-parabolic equation with nonstandard growth conditions, Appl. Anal., (2021). doi: 10.1080/00036811.2020.1869941
    [24] Blow-up and extinction for a thin-film equation with initial-boundary value conditions. J. Math. Anal. Appl. (2016) 436: 796-809.
    [25] Mathematical modeling of electrorheological materials. Continu. Mech. Thermodyn. (2001) 13: 59-78.
    [26] On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. (1968) 30: 148-172.
    [27] Nonlinear degenerate evolution equations and partial differential equations of mixed type. SIAM J. Math. Anal. (1975) 6: 25-42.
    [28] On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete Contin. Dyn. Syst. Ser. B (2021) 26: 5465-5494.
    [29] Kirchhoff-type system with linear weak damping and logarithmic nonlinearities. Nonlinear Anal. (2019) 188: 475-499.
    [30] Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv. Nonlinear Anal. (2021) 10: 261-288.
    [31] Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity (2018) 31: 3228-3250.
    [32] Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. (2013) 264: 2732-2763.
    [33] Second critical exponent and life span for pseudo-parabolic equation. J. Differential Equations (2012) 253: 3286-3303.
    [34] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675–710,877.
    [35] On Lavrentiev's phenomenon. Russ. J. Math. Phys. (1995) 3: 249-269.
    [36] Solvability of the three-dimensional thermistor problem. Proc. Stekolov Inst. Math. (2008) 261: 98-111.
    [37] Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth. Nonlinear Anal. Real World Appl. (2019) 48: 54-70.
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