Opuscula Math. 42, no. 2 (2022), 305-335
https://doi.org/10.7494/OpMath.2022.42.2.305

 
Opuscula Mathematica

On some inverse problem for bi-parabolic equation with observed data in Lp spaces

Nguyen Huy Tuan

Abstract. The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in \(L^p\). We are interested in looking at three types of inverse problems. Regularization results in the \(L^2\) space appears in many related papers, but the survey results are rare in \(L^p\), \(p \neq 2\). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in \(L^p\) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in \(L^p\), we obtain the approximated solution also in the space \(L^p\). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space \(L^p\). This paper seems to generalize to previous results for bi-parabolic equation on this direction.

Keywords: bi-parabolic equations, Fourier truncation method, inverse source parabolic, inverse initial problem, Sobolev embeddings, Sobolev embeddings.

Mathematics Subject Classification: 35A05, 35A08.

Full text (pdf)

  1. K. Atifi, E.H. Essoufi, B. Khouiti, An inverse backward problem for degenerate two-dimensional parabolic equation, Opuscula Math. 40 (2020), no. 4, 427-449.
  2. M.K. Beshtokov, Local and nonlocal boundary value problems for degenerating and nondegenerating pseudoparabolic equations with a Riemann-Liouville fractional derivative, Differ. Equ. 54 (2018), 758-774, Translation of Differ. Uravn. 54 (2018), no. 6, 763-778.
  3. K. Besma, B. Nadji, R. Faouzia, A modified quasi-boundary value method for an abstract ill-posed biparabolic problem, Open Math. 15 (2017), 1649-1666.
  4. V.M. Bulavatsky, Mathematical modeling of filtrational consolidation of soil under motion of saline solutions on the basis of biparabolic model, J. Autom. Inform. Sci. 35 (2003), no. 8, 13-22.
  5. V.M. Bulavatsky, Fractional differential analog of biparabolic evolution equation and some its applications, Cybern. Syst. Anal. 52 (2016), no. 5, 337-347.
  6. V.M. Bulavatsky, V.V. Skopetsky, Generalized mathematical model of the dynamics of consolidation processes with relaxation, Cybern. Syst. Anal. 44 (2008), no. 5, 646-654.
  7. N. Chafee, Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions, J. Differential Equations 18 (1975), no. 1, 111-134.
  8. C.M. Dafermos, M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Analysis 13 (1973), no. 1, 97-106.
  9. M. Ebenbeck, F.K. Lam, Weak and stationary solutions to a Cahn-Hilliard-Brinkman model with singular potentials and source terms, Adv. Nonlinear Anal. 10 (2021), no. 1, 24-65.
  10. V.L. Fushchich, A.S. Galitsyn, A.S. Polubinskii, A new mathematical model of heat conduction processes, Ukrainian Math. J. 42 (1990), 210-216.
  11. T.E. Ghoul, N.V. Tien, H. Zaag, Construction of type I blowup solutions for a higher order semilinear parabolic equation, Adv. Nonlinear Anal. 9 (2020), no. 1, 388-412.
  12. C.H. Grunau, N. Miyake, S. Okabe, Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations, Adv. Nonlinear Anal. 10 (2021), no. 1, 353-370.
  13. B.J. Greer, A.L. Bertozzi, G. Sapiro, Fourth order partial differential equations on general geometries, J. Comput. Phys. 216 (2006), no. 1, 216-246.
  14. L. Joseph, D.D. Preziosi, Heat waves, Rev. Mod. Physics (1989), 41-73.
  15. V. Kalantarov, S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations 247 (2009), no. 4, 1120-1155.
  16. A. Lakhdari, N. Boussetila, An iterative regularization method for an abstract ill-posed biparabolic problem, Bound. Value Probl. 55 (2015), 1-17.
  17. N.H. Luc, L.D. Long, H.T.K. Van, V.T. Nguyen, A nonlinear fractional Rayleigh-Stokes equation under nonlocal integral conditions, Advances in Difference Equations (2021), to appear.
  18. M. Marras, S.V. Piro, Bounds for blow-up time in nonlinear parabolic systems, Disctete Contin. Dyn. Syst. Ser. A (2011), 1025-1031.
  19. D.H.Q. Nam, L.D. Long, D. O'Regan, T.B. Ngoc, N.H. Tuan, Identification of the right-hand side in a bi-parabolic equation with final data, Applicable Analysis, to appear.
  20. V. Pata, M. Squassina, On the strongly damped wave equation, Comm. Math. Phys. 253 (2005), no. 3, 511-533.
  21. L.E. Payne, On a proposed model for heat conduction, IMA J. Appl. Math. 71 (2006), 590-599.
  22. N.D. Phuong, N.H. Luc, L.D. Long, Modified quasi boundary value method for inverse source problem of the bi-parabolic equation, Advances in the Theory of Nonlinear Analysis and its Applications 4 (2020), no. 3, 132-142.
  23. A. Segatti, J.L. Vázquez, On a fractional thin film equation, Adv. Nonlinear Anal. 9 (2020), no. 1, 1516-1558.
  24. N.H. Tuan, T. Caraballo, On initial and terminal value problems for fractional nonclassical diffusion equations, Proc. Amer. Math. Soc. 149 (2021), no. 1, 143-161.
  25. N.H. Tuan, T. Caraballo, T.N. Thach, On terminal value problems for bi-parabolic equations driven by Wiener process and fractional Brownian motions, Asymptot. Anal. 123 (2021), no. 3-4, 335-366.
  26. N.H. Tuan, V.V. Au, R. Xu, R. Wang, On the initial and terminal value problem for a class of semilinear strongly material damped plate equations, J. Math. Anal. Appl. 492 (2020), no. 2, 124481.
  27. N.H. Tuan, V.A. Khoa, V.V. Au, Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements, SIAM J. Math. Anal. 51 (2019), no. 1, 60-85.
  28. N.H. Tuan, M. Kirane, D.H.Q. Nam, V.V. Au, Approximation of an inverse initial problem for a biparabolic equation, Mediterr. J. Math. 15 (2018), Article no. 18.
  29. H.T.K. Van, Note on abstract elliptic equations with nonlocal boundary in time condition, Advances in the Theory of Nonlinear Analysis and its Applications 5 (2021), no. 4, 551-558.
  30. L. Wang, X. Zhou, X. Wei, Heat Conduction: Mathematical Models and Analytical Solutions, Springer, Berlin, 2008.
  31. Y. Yang, Md Salik Ahmed, L. Qin, R. Xu, Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations, Opuscula Math. 39 (2019), no. 2, 297-313.
  • Nguyen Huy Tuan
  • Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
  • Communicated by Ruzhang Xu.
  • Received: 2021-10-07.
  • Revised: 2022-01-05.
  • Accepted: 2022-01-08.
  • Published online: 2022-02-25.
Opuscula Mathematica - cover

Cite this article as:
Nguyen Huy Tuan, On some inverse problem for bi-parabolic equation with observed data in Lp spaces, Opuscula Math. 42, no. 2 (2022), 305-335, https://doi.org/10.7494/OpMath.2022.42.2.305

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.