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

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.
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.  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, (2003).  doi: 10.1201/9780203911433.  Google Scholar

[3]

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

[5]

V. E. Fedorov and M. V. Plekhanova, Optimal control of Sobolev type linear equations,, Differ. Equ., 40 (2004), 1627.  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.  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.  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], (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.  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.   Google Scholar

[11]

N. Sidorov, B. Loginov, A. Sinitsyn and M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications,, Dordrecht, (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], (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.   Google Scholar

[14]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators,, Utrecht, (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.  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.   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.  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, (2003).  doi: 10.1201/9780203911433.  Google Scholar

[3]

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

[5]

V. E. Fedorov and M. V. Plekhanova, Optimal control of Sobolev type linear equations,, Differ. Equ., 40 (2004), 1627.  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.  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.  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], (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.  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.   Google Scholar

[11]

N. Sidorov, B. Loginov, A. Sinitsyn and M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications,, Dordrecht, (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], (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.   Google Scholar

[14]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators,, Utrecht, (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.  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.   Google Scholar

[1]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[2]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[3]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[4]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[5]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[6]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[7]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[8]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171

[9]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[10]

Helin Guo, Huan-Song Zhou. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1023-1050. doi: 10.3934/dcds.2020308

[11]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[12]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[13]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[14]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, 2021, 14 (1) : 149-174. doi: 10.3934/krm.2020052

[15]

Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380

[16]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[17]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[18]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[19]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

[20]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]