Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity
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 v...
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Ivan Franko National University of Lviv
2023-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/396 |
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| author | M. M. Bokalo O. V. Domanska |
| author_facet | M. M. Bokalo O. V. Domanska |
| author_sort | M. M. Bokalo |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-4e7909f2e67e499c856a9dba469524e4 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2023-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-4e7909f2e67e499c856a9dba469524e42025-08-20T03:33:11ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202023-03-015918610510.30970/ms.59.1.86-105396Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinityM. M. Bokalo0O. V. Domanska1Ivan Franko National University of Lviv, LvivDepartment of Mechanics and Mathematics Ivan Franko National University of Lviv Lviv, UkraineInitial-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.http://matstud.org.ua/ojs/index.php/matstud/article/view/396initial-boundary value problemhigher-order elliptic-parabolic equationnonlinearelliptic-parabolic equationweak solution |
| spellingShingle | M. M. Bokalo O. V. Domanska Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity Математичні Студії initial-boundary value problem higher-order elliptic-parabolic equation non linear elliptic-parabolic equation weak solution |
| title | Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity |
| title_full | Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity |
| title_fullStr | Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity |
| title_full_unstemmed | Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity |
| title_short | Initial-boundary value problem for higher-orders nonlinear elliptic-parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity |
| title_sort | initial boundary value problem for higher orders nonlinear elliptic parabolic equations with variable exponents of the nonlinearity in unbounded domains without conditions at infinity |
| topic | initial-boundary value problem higher-order elliptic-parabolic equation non linear elliptic-parabolic equation weak solution |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/396 |
| work_keys_str_mv | AT mmbokalo initialboundaryvalueproblemforhigherordersnonlinearellipticparabolicequationswithvariableexponentsofthenonlinearityinunboundeddomainswithoutconditionsatinfinity AT ovdomanska initialboundaryvalueproblemforhigherordersnonlinearellipticparabolicequationswithvariableexponentsofthenonlinearityinunboundeddomainswithoutconditionsatinfinity |