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June  2016, 9(3): 833-846. doi: 10.3934/dcdss.2016031

Strong solutions of quasilinear equations in Banach spaces not solvable with respect to the highest-order derivative

Marina V. Plekhanova 1,

1. 

South Ural State University, 76 Lenina Av., Chelyabinsk, 454080, Russian Federation

Received  March 2015 Revised  June 2015 Published  April 2016

By means of the Mittag-Leffler function existence and uniqueness conditions are obtained for a strong solution of the Cauchy problem to quasilinear differential equation in a Banach space, solved with respect to the highest-order derivative. The results are used in the study of quasilinear equations with degenerate operator at the highest-order derivative. Some special restrictions for nonlinear operator in the equation are used here. Existence conditions of a unique strong solution for the Cauchy problem and generalized Showalter--Sidorov for degenerate quasilinear equations were found. The obtained results are illustrated by an example of initial-boundary value problem for a quasilinear system of equations not resolved with respect to the highest-order time derivative.
Keywords: Cauchy problem, generalized Showalter--Sidorov problem, strong solution, degenerate evolution equation, quasilinear equation, initial boundary value problem., High order equation.
Mathematics Subject Classification: Primary: 34G20; Secondary: 47J35, 35G61, 35D3.
Citation: Marina V. Plekhanova. Strong solutions of quasilinear equations in Banach spaces not solvable with respect to the highest-order derivative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 833-846. doi: 10.3934/dcdss.2016031
References:
[1]

P. N. Davydov and V. E. Fedorov, On Nonlocal Solutions of Semilinear Equations of the Sobolev Type, Differ. Equ., 49 (2013), 326-335. doi: 10.1134/S0012266113030087.  Google Scholar

[2]

G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York, Basel, 2003. doi: 10.1201/9780203911433.  Google Scholar

[3]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York, 1999.  Google Scholar

[4]

V. E. Fedorov and P. N. Davydov, Polulinejnye vyrozhdennye evolyucionnye uravneniya i nelinejnye sistemy gidrodinamicheskogo tipa, (Russian) [Degenerate semilinear evolution equations and nonlinear systems of hydrodynamic type] Trudy instituta matematiki i mekhaniki UrO RAN, 19 (2013), 267-278.  Google Scholar

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V. E. Fedorov and M. V. Plekhanova, Optimal control of Sobolev type linear equations, Differ. Equ., 40 (2004), 1627-1637. doi: 10.1007/s10625-005-0082-9.  Google Scholar

[6]

A. I. Kozhanov, Boundary value problems for some classes of higher-order equations that are unsolved with respect to the highest derivative, Siberian Math. J., 35 (1994), 359-376. doi: 10.1007/BF02104779.  Google Scholar

[7]

M. V. Plekhanova and V. E. Fedorov, On the existence and uniqueness of solutions of optimal control problems of linear distributed systems which are not solved with respect to the time derivative, Izvestiya: Mathematics, 75 (2011), 395-412. doi: 10.1070/IM2011v075n02ABEH002538.  Google Scholar

[8]

M. V. Plekhanova and V. E. Fedorov, Optimal'noe Upravlenie Vyrozddennymi Raspredelennymi Sistemami, (Russian) [Optimal control of degenerate distributed systems], South Ural State University, Chelyabinsk, 2013. Google Scholar

[9]

R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1975), 25-42. doi: 10.1137/0506004.  Google Scholar

[10]

N. A. Sidorov, Ob odnom klasse vyrozhdennyx differencialnyx uravneniy s konvergentsiey, (Russian) [On a class of degenerate differential equations with convergence] Mathematical Notes, 63 (1984), 569-578. Google Scholar

[11]

N. Sidorov, B. Loginov, A. Sinitsyn and M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications, Dordrecht, Boston, London, 2002. doi: 10.1007/978-94-017-2122-6.  Google Scholar

[12]

A. G. Sveshnikov, A. B. Al'shin, M. O. Korpusov and Yu. D. Pletner, Lineinye I Nelineinye Uravneniya Sobolevskogo Tipa, (Russian) [Linear and Nonlinear Equations of the Sobolev Type], Fizmatlit, Moscow, 2007. Google Scholar

[13]

G. A. Sviridyuk, Polulinejnye uravneniya tipa soboleva s otnositelno ogranichennym operatorom, (Russian) [Semilinear equations of Sobolev type with relatively bounded operator] Doklady AN SSSR, 318 (1991), 828-831. Google Scholar

[14]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Utrecht, Boston, 2003. doi: 10.1515/9783110915501.  Google Scholar

[15]

G. A. Sviridyuk and T. G. Sukacheva, On the solvability of a nonstationary problem in the dynamics of an incompressible viscoelastic fluid, Mathematical Notes, 63 (1998), 388-395. doi: 10.1007/BF02317787.  Google Scholar

[16]

A. A. Zamyshlyaeva and E. V. Bychkov, Fazovoe prostranstvo modifitsirovannogo uravneniya Bussineska, (Russian) [The phase space of the modified Boussinesq equation] Vestnik of South Ural State University. Ser. Matematicheskoe modelirovanie i programmirovanie, 19, (2012), 13-19. Google Scholar

show all references

References:
[1]

P. N. Davydov and V. E. Fedorov, On Nonlocal Solutions of Semilinear Equations of the Sobolev Type, Differ. Equ., 49 (2013), 326-335. doi: 10.1134/S0012266113030087.  Google Scholar

[2]

G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York, Basel, 2003. doi: 10.1201/9780203911433.  Google Scholar

[3]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York, 1999.  Google Scholar

[4]

V. E. Fedorov and P. N. Davydov, Polulinejnye vyrozhdennye evolyucionnye uravneniya i nelinejnye sistemy gidrodinamicheskogo tipa, (Russian) [Degenerate semilinear evolution equations and nonlinear systems of hydrodynamic type] Trudy instituta matematiki i mekhaniki UrO RAN, 19 (2013), 267-278.  Google Scholar

[5]

V. E. Fedorov and M. V. Plekhanova, Optimal control of Sobolev type linear equations, Differ. Equ., 40 (2004), 1627-1637. doi: 10.1007/s10625-005-0082-9.  Google Scholar

[6]

A. I. Kozhanov, Boundary value problems for some classes of higher-order equations that are unsolved with respect to the highest derivative, Siberian Math. J., 35 (1994), 359-376. doi: 10.1007/BF02104779.  Google Scholar

[7]

M. V. Plekhanova and V. E. Fedorov, On the existence and uniqueness of solutions of optimal control problems of linear distributed systems which are not solved with respect to the time derivative, Izvestiya: Mathematics, 75 (2011), 395-412. doi: 10.1070/IM2011v075n02ABEH002538.  Google Scholar

[8]

M. V. Plekhanova and V. E. Fedorov, Optimal'noe Upravlenie Vyrozddennymi Raspredelennymi Sistemami, (Russian) [Optimal control of degenerate distributed systems], South Ural State University, Chelyabinsk, 2013. Google Scholar

[9]

R. E. Showalter, Nonlinear degenerate evolution equations and partial differential equations of mixed type, SIAM J. Math. Anal., 6 (1975), 25-42. doi: 10.1137/0506004.  Google Scholar

[10]

N. A. Sidorov, Ob odnom klasse vyrozhdennyx differencialnyx uravneniy s konvergentsiey, (Russian) [On a class of degenerate differential equations with convergence] Mathematical Notes, 63 (1984), 569-578. Google Scholar

[11]

N. Sidorov, B. Loginov, A. Sinitsyn and M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications, Dordrecht, Boston, London, 2002. doi: 10.1007/978-94-017-2122-6.  Google Scholar

[12]

A. G. Sveshnikov, A. B. Al'shin, M. O. Korpusov and Yu. D. Pletner, Lineinye I Nelineinye Uravneniya Sobolevskogo Tipa, (Russian) [Linear and Nonlinear Equations of the Sobolev Type], Fizmatlit, Moscow, 2007. Google Scholar

[13]

G. A. Sviridyuk, Polulinejnye uravneniya tipa soboleva s otnositelno ogranichennym operatorom, (Russian) [Semilinear equations of Sobolev type with relatively bounded operator] Doklady AN SSSR, 318 (1991), 828-831. Google Scholar

[14]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Utrecht, Boston, 2003. doi: 10.1515/9783110915501.  Google Scholar

[15]

G. A. Sviridyuk and T. G. Sukacheva, On the solvability of a nonstationary problem in the dynamics of an incompressible viscoelastic fluid, Mathematical Notes, 63 (1998), 388-395. doi: 10.1007/BF02317787.  Google Scholar

[16]

A. A. Zamyshlyaeva and E. V. Bychkov, Fazovoe prostranstvo modifitsirovannogo uravneniya Bussineska, (Russian) [The phase space of the modified Boussinesq equation] Vestnik of South Ural State University. Ser. Matematicheskoe modelirovanie i programmirovanie, 19, (2012), 13-19. Google Scholar

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