Nonstationary heat equation with nonlinear side condition
Abstract
The initial boundary value problem for the nonstationary heat equation is studied in a bounded domain with the specific overdetermination condition. This condition is nonlinear and can be interpreted as the energy functional. In present paper we construct the class of solutions to this problem.
Keyword : nonstationary heat equation, inverse problem, very weak solution, nonlinear side condition
How to Cite
Belickas, T., Kaulakytė, K., & Puriuškis, G. (2025). Nonstationary heat equation with nonlinear side condition. Mathematical Modelling and Analysis, 30(1), 109–119. https://doi.org/10.3846/mma.2025.20204
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
R.A. Adams. Sobolev Spaces. Academic Press, New York, San Francisco, London, 1975.
P. Agarwal, J. Merker and G. Schuldt. Singular integral Neumann boundary conditions for semilinear elliptic PDEs. Axioms, 10(2), 2021. https://doi.org/10.3390/axioms10020074
J.R. Cannon. The solution of the heat equation subject to specification of energy. Quart. Appl. Math., 21:155–160, 1963. https://doi.org/10.1090/QAM/160437
J.R. Cannon. The One-Dimensional Heat Equation. Cambridge University Press, Addison-Wesley, Menoro Park, 1984. https://doi.org/10.1017/CBO9781139086967
L.C. Evans. Partial Differential Equations. American Mathematical Society, 2010.
G.P. Galdi, K. Pileckas and A. Silvestre. On the unsteady Poiseuille flow in a pipe. Zeitschrift fur angew. Mathematik und Physik, 58:994–1007, 2007. https://doi.org/10.1007/s00033-006-6114-3
K. Kaulakytė, N. Kozulinas and K. Pileckas. Time-periodic Poiseuille-type solution with minimally regular flow-rate. Nonlinear Analysis: Modelling and Control, 26(5):947–968, 2021. https://doi.org/10.15388/namc.2021.26.24502
O.A. Ladyzhenskaya. The Boundary Value Problems of Mathematical Physics. Springer New York, NY, 1985. https://doi.org/10.1007/978-1-4757-4317-3
J. Merker and A. Matas. Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem. Partial Differential Equations in Applied Mathematics, 5:100384, 2022. https://doi.org/10.1016/j.padiff.2022.100384
K. Pileckas. Existence of solutions with the prescribed flux of the Navier– Stokes system in an infinite cylinder. J. Math. Fluid Mech., 8(4):542–563, 2006. https://doi.org/10.1007/s00021-005-0187-1
K. Pileckas. Navier–Stokes system in domains with cylindrical outlets to infinity Leray’s problem. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 4, pp. 445–647. Elsevier, Amsterdam, 2007. https://doi.org/10.1016/S1874-5792(07)80012-7
K. Pileckas. Solvability in weighted spaces of the three-dimensional Navier– Stokes problem in domains with cylindrical outlets to infinity. Topol. Methods Nonlinear Anal., 29(2):333–360, 2007.
K. Pileckas. On the behavior of the nonstationary Poiseuille solution as t →∞. Siberian Math.J.,46:707–716,2005. https://doi.org/10.1007/s11202-005-0071-5
K. Pileckas and V. Keblikas. On the existence of the nonstationary Poiseuille solution. Siberian Math. J., 46(3):514–526, 2005. https://doi.org/10.1007/s11202-005-0053-7
K. Pileckas and R. Ciegis. Existence of nonstationary Poiseuille type solutionsˇ under minimal regularity assumptions. Z. Angew. Math. Phys., 71(192), 2020. https://doi.org/10.1007/s00033-020-01422-5
S.S. Sritharan. On the acceleration of viscous fluid through an unbounded channel. J. Math. Anal. Appl., 168(1):255–283, 1992. https://doi.org/10.1016/0022-247X(92)90204-Q
P. Agarwal, J. Merker and G. Schuldt. Singular integral Neumann boundary conditions for semilinear elliptic PDEs. Axioms, 10(2), 2021. https://doi.org/10.3390/axioms10020074
J.R. Cannon. The solution of the heat equation subject to specification of energy. Quart. Appl. Math., 21:155–160, 1963. https://doi.org/10.1090/QAM/160437
J.R. Cannon. The One-Dimensional Heat Equation. Cambridge University Press, Addison-Wesley, Menoro Park, 1984. https://doi.org/10.1017/CBO9781139086967
L.C. Evans. Partial Differential Equations. American Mathematical Society, 2010.
G.P. Galdi, K. Pileckas and A. Silvestre. On the unsteady Poiseuille flow in a pipe. Zeitschrift fur angew. Mathematik und Physik, 58:994–1007, 2007. https://doi.org/10.1007/s00033-006-6114-3
K. Kaulakytė, N. Kozulinas and K. Pileckas. Time-periodic Poiseuille-type solution with minimally regular flow-rate. Nonlinear Analysis: Modelling and Control, 26(5):947–968, 2021. https://doi.org/10.15388/namc.2021.26.24502
O.A. Ladyzhenskaya. The Boundary Value Problems of Mathematical Physics. Springer New York, NY, 1985. https://doi.org/10.1007/978-1-4757-4317-3
J. Merker and A. Matas. Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem. Partial Differential Equations in Applied Mathematics, 5:100384, 2022. https://doi.org/10.1016/j.padiff.2022.100384
K. Pileckas. Existence of solutions with the prescribed flux of the Navier– Stokes system in an infinite cylinder. J. Math. Fluid Mech., 8(4):542–563, 2006. https://doi.org/10.1007/s00021-005-0187-1
K. Pileckas. Navier–Stokes system in domains with cylindrical outlets to infinity Leray’s problem. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 4, pp. 445–647. Elsevier, Amsterdam, 2007. https://doi.org/10.1016/S1874-5792(07)80012-7
K. Pileckas. Solvability in weighted spaces of the three-dimensional Navier– Stokes problem in domains with cylindrical outlets to infinity. Topol. Methods Nonlinear Anal., 29(2):333–360, 2007.
K. Pileckas. On the behavior of the nonstationary Poiseuille solution as t →∞. Siberian Math.J.,46:707–716,2005. https://doi.org/10.1007/s11202-005-0071-5
K. Pileckas and V. Keblikas. On the existence of the nonstationary Poiseuille solution. Siberian Math. J., 46(3):514–526, 2005. https://doi.org/10.1007/s11202-005-0053-7
K. Pileckas and R. Ciegis. Existence of nonstationary Poiseuille type solutionsˇ under minimal regularity assumptions. Z. Angew. Math. Phys., 71(192), 2020. https://doi.org/10.1007/s00033-020-01422-5
S.S. Sritharan. On the acceleration of viscous fluid through an unbounded channel. J. Math. Anal. Appl., 168(1):255–283, 1992. https://doi.org/10.1016/0022-247X(92)90204-Q