In this paper, we investigate a final boundary value problem for a class of fractional with parameter $ \beta $ pseudo-parabolic partial differential equations with nonlinear reaction term. For $ 0<\beta < 1, $ the solution is regularity-loss, we establish the well-posedness of solutions. In the case that $ \beta >1 $, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.
Citation: Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation[J]. Electronic Research Archive, 2021, 29(1): 1709-1734. doi: 10.3934/era.2020088
In this paper, we investigate a final boundary value problem for a class of fractional with parameter $ \beta $ pseudo-parabolic partial differential equations with nonlinear reaction term. For $ 0<\beta < 1, $ the solution is regularity-loss, we establish the well-posedness of solutions. In the case that $ \beta >1 $, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.
[1] | On a class of fully nonlinear parabolic equations. Adv. Nonlinear Anal. (2019) 8: 79-100. |
[2] | Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete Contin. Dyn. Syst. (2019) 39: 771-801. |
[3] | Identification of the initial condition in backward problem with nonlinear diffusion and reaction. J. Math. Anal. Appl. (2017) 452: 176-187. |
[4] | Initial boundary value problem for a mixed pseudo- parabolic $p$-Laplacian type equation with logarithmic nonlinearity. Electronic J. Differential Equations (2018) 2018: 1-19. |
[5] | Cauchy problems of semilinear pseudo-parabolic equations. J. Differential Equations (2009) 246: 4568-4590. |
[6] | Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differential Equations (2015) 258: 4424-4442. |
[7] | Global existence and blow-up in finite time for a class of finitely degenerate semilinear pseudo-parabolic equations. Acta Math. Sin. (Engl. Ser.) (2019) 35: 1143-1162. |
[8] | 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. |
[9] | Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete Contin. Dyn. Syst. Ser. B (2016) 21: 781-801. |
[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] | Solutions of pseudo-heat equations in the whole space. Arch. Ration. Mech. Anal. (1972/73) 49: 57-78. |
[12] | Blow-up and decay for a class of pseudo-parabolic $p$-Laplacian equation with logarithmic nonlinearity. Comput. Math. Appl. (2018) 75: 459-469. |
[13] | Regularity of solutions of the parabolic normalized $p$-Laplace equation. Adv. Nonlinear Anal. (2020) 9: 7-15. |
[14] | The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation. Comput. Math. Appl. (2017) 73: 2221-2232. |
[15] | Well-posed final value problems and Duhamel's formula for coercive Lax-Milgram operators. Electronic Res. Arch. (2019) 27: 20-36. |
[16] | M. V. Klibanov, Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems, 31 (2015), 20pp. doi: 10.1088/0266-5611/31/12/125007 |
[17] | Global existence and blow up of solutions for pseudo-parabolic equation with singular potential. J. Differential Equations (2020) 269: 4914-4959. |
[18] | Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. (2020) 9: 613-632. |
[19] | Y. Lu and L. Fei, Bounds for blow-up time in a semilinear pseudo-parabolic equation with nonlocal source, J. Inequal. Appl., 2016 (2016), 11pp. doi: 10.1186/s13660-016-1171-4 |
[20] | Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements. SIAM J. Math. Anal. (2019) 51: 60-85. |
[21] | The well-posedness and regularity of a rotating blades equation. Electron. Res. Arch. (2020) 28: 691-719. |
[22] | Pseudoparabolic partial differential equations. SIAM J. Math. Anal. (1970) 1: 1-26. |
[23] | Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term. Appl. Anal. (2019) 98: 735-755. |
[24] | Parabolic and pseudo-parabolic partial differential equations. J. Math. Soc. Japan (1969) 21: 440-453. |
[25] | N. H. Tuan, V. V. Au, V. A. Khoa and D. Lesnic, Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction, Inverse Problems, 33 (2017), 40pp. doi: 10.1088/1361-6420/aa635f |
[26] | A new fourier truncated regularization method for semilinear backward parabolic problems. Acta Appl. Math. (2017) 148: 143-155. |
[27] | A nonlinear parabolic equation backward in time: Regularization with new error estimates. Nonlinear Anal. (2010) 73: 1842-1852. |
[28] | R. Wang, Y. Li and B. Wang, Bi-spatial pullback attractors of fractional nonclassical diffusion equations on unbounded domains with $(p, q)$-growth nonlinearities, Appl. Math. Optim., (2020). doi: 10.1007/s00245-019-09650-6 |
[29] | Random dynamics of fractional nonclassical diffusion equations driven by colored noise. Discrete Contin. Dyn. Syst. (2019) 39: 4091-4126. |
[30] | Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb R^N$. Nonlinearity (2019) 32: 4524-4556. |
[31] | Global well-posedness of coupled parabolic systems. Sci. China Math. (2020) 63: 321-356. |
[32] | Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. (2013) 264: 2732-2763. |
[33] | Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy. Appl. Math. Lett. (2018) 83: 176-181. |
[34] | The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete Contin. Dyn. Syst. (2017) 37: 5631-5649. |
[35] | Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation. Comput. Math. Appl. (2014) 68: 1787-1793. |
[36] | Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term. Appl. Math. Comput. (2018) 329: 38-51. |
[37] | A sufficient condition for blowup of solutions to a class of pseudo-parabolic equations with a nonlocal term. Math. Methods Appl. Sci. (2016) 39: 3591-3606. |