Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity
Abstract
Initial-boundary value problems for parabolic and elliptic-parabolic (that is degenerated parabolic) equations in unbounded domains with respect to the spatial variables were studied by many authors. It is well known that in order to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic and elliptic-parabolic equations in unbounded domains we need some restrictions on behavior of solution as $|x|\to +\infty$ (for example, growth restriction of solution as $|x|\to +\infty$, or the solution to belong to some functional spaces).
Note, that we need some restrictions on the data-in behavior as $|x|\to +\infty$ for the initial-boundary value problems for equations considered above to be solvable.
However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are uniquely solvable without any conditions at infinity.
We prove the unique solvability of the initial-boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. A priori estimate of the weak solutions of this problem was also obtained. As far as we know, the initial-boundary value problem for the higher-orders anisotropic elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains were not considered before.
References
F. Bernis, Elliptic and parabolic semilinear parabolic problems without conditions at infinity, Arch. Rational Mech. Anal., 106 (1989), №3, 217–241.
Ph. Benilan, M.G. Grandall, M. Pierre, Solutions of the porous medium equations in $mathbb R^n$ under optimal conditions on initial values, Indiana Univ. Math. J., 33 (1984), №1, 51–87.
L. Boccardo, Th. Gallou¨et, J.L. Vazquez, Solutions of nonlinear parabolic equations without growth restrictions on the data, Electronic J. Diff. Eq., 60 (2001), 1–20.
M.M. Bokalo, Boundary value problems for semilinear parabolic equations in unbounded domains without conditions at infinity, Siberian Math. J., 37 (1996), №5, 860–867.
N.M. Bokalo, The well-posedness of the first boundary value problem and the Cauchy problem for some quasilinear parabolic systems without conditions at infinity, J. Math. Sci., 135 (2006), №1, 2625–2636.
M.M. Bokalo, I.B. Pauchok, On the well-posedness of a Fourier problem for nonlinear parabolic equations of higher order with variable exponents of nonlinearity, Mat. Stud., 26 (2006), №1, 25–48.
M.M. Bokalo, O.M. Buhrii, R.A. Mashiyev, Unique solvability of initial-boundary-value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, J. Nonl. Evol. Eq. Appl., 6 (2013), 67–87.
M. Bokalo, O. Buhrii, N. Hryadil, Initial-boundary value problems for nonlinear elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains without conditions at infinity, Nonlinear Analysis. Elsevier. USA, 192 (2020), 1–17.
M. Bokalo, Initial-boundary value problems for anisotropic parabolic equations with variable exponents of the nonlinearity in unbounded domains with conditions at infinity, Journal of optimization, differential equations and their applications (JODEA), 30 (2022) №1, 98–121. doi 10.15421/142205.
M. Bokalo, Initial-boundary value problem for higher-orders nonlinear parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity, Bukovinian Math. Journal, 10 (2022), №2, 59–76.
H. Brezis, Semilinear equations in $mathbb{R}^N$ without conditions at infinity, Appl. Math. Optim., 12 (1984), №3, 271–282.
O. Buhrii, N. Buhrii, Nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities, J. Math. Anal. Appl., 473 (2019), 695–711.
L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents.Springer, Heidelberg, 2011.
A. Gladkov, M. Guedda, Diffusion-absorption equation without growth restrictions on the data at infinity, J. Math. Anal. Appl., 274 (2002), №1, 16–37.
O. Kovacik, J. Rakosnic, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), №116, 592–618.
O. Kovacik, Parabolic equations in generalized Sobolev spaces $W^{k,p(x)}$, Fasciculi Mathematici, 25 (1995), 87–94.
J.-L. Lions, Quelques m´ethodes de r´esolution des probl´emes aux limites non lin´eaires, Paris, France: Dunod Gauthier-Villars, 1969.
R.A. Mashiyev, O.M. Buhrii, Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity. Journal of Mathematical Analysis and Applications, 377 (2011), 450–463.
C. Marchi, A. Tesei, Higher-order parabolic equations without conditions at infinity, J. Math. Anal. Appl., 269 (2002), 352–368.
O.A. Oleinik, G.A. Iosifyan, An analog of Saint-Venant principle and uniqueness of the solutions of the boundary-value problems in unbounded domains for parabolic equations. Usp. Mat. Nauk, 31 (1976), №6, 142–166. (in Russian)
V. Radulescu, D. Repovs, Partial differential equations with variable exponents: variational methods and qualitative analysis, CRC Press, Boca Raton, London, New York, 2015.
M. Ruzicka, Electroreological fluids: modeling and mathematical theory, Springer-Verl., Berlin, 2000.
V.N. Samokhin, On a class of equations that generalize equations of polytropic filtration, Diff. Equat., 32 (1996), №5, 648–657. (in Russian)
A.N. Tikhonov, Th´eoremes dunicite pour lequation de la chaleur, Mat. Sb., 42 (1935), №2, 199–216.
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