This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.
References
[1]
Pao, C.V. (2012) Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York.
[2]
Hu, B. (2011) Blow-Up Theories for Semilinear Parabolic Equations. Springer, Berlin. https://doi.org/10.1007/978-3-642-18460-4
[3]
Levine, H.A. (1990) The Role of Critical Exponents in Blowup Theorems. SIAM Review, 32, 262-288. https://doi.org/10.1137/1032046
[4]
Galaktionov, V.A. and Vazquez, J.-L. (2002) The Problem of Blow-Up in Nonlinear Parabolic Equations. Discrete and Continuous Dynamical Systems, 8, 399-434. https://doi.org/10.3934/dcds.2002.8.399
[5]
El-Gamel, M., El-Baghdady, G.I. and El-Hady, M.A. (2022) Highly Efficient Method for Solving Parabolic PDE with Nonlocal Boundary Conditions. Applied Mathematics, 13, 101-119. https://doi.org/10.4236/am.2022.132009
[6]
Sheng, L.X, Hu, W.M., Su, Y.H. and Yun, Y.Z. (2023) Existence of Solutions for a Non-Autonomous Evolution Equations with Nonlocal Conditions. Journal of Applied Mathematics and Physics, 11, 4079-4091. https://doi.org/10.4236/jamp.2023.1112260
[7]
Yan, L.J. and Yang, Z.D. (2018) Blow-Up and Non-Extinction for a Nonlocal Parabolic Equation with Logarithmic Nonlinearity. Boundary Value Problems, 2018, Article No. 121. https://doi.org/10.1186/s13661-018-1042-7
[8]
Le, C.N. and Le, X.T. (2017) Global Solution and Blow-Up for a Class of Pseudo-Laplacian Evolution Equations with Logarithmic Nonlinearity. Computers and Mathematics with Applications, 151, 149-169. https://doi.org/10.1007/s10440-017-0106-5
[9]
Hardy, G.H. and Littlewood, J.E. and Polya, G. (1934) Inequalities (Cambridge Mathematical Library). Subsequent Edition, Cambridge University Press, Cambridge.
[10]
Do, T.D., Trong, N.N. and Thanh, B.L.T. (2023) On a Higher-Order Reaction-Diffusion Equation with a Special Medium Void via Potential Well Method. Taiwanese Journal of Mathematics, 27, 53-79. https://doi.org/10.11650/tjm/220703
[11]
Han, Y.Z. (2018) A New Blow-Up Criterion for Non-Newton Filtration Equations with Special Medium Void. Rocky Mountain Journal of Mathematics, 48, 2489-2501. https://doi.org/10.1216/RMJ-2018-48-8-2489
[12]
Tan, Z. (2002) The Reaction-Diffusion Equation with Lewis Function and Critical Sobolev Exponent. Journal of Mathematical Analysis and Applications, 272, 480-495. https://doi.org/10.1016/S0022-247X(02)00166-X
