The Destruction of the Solution of the Nonlocal Equation with Gradient Nonlinearity

In this paper we continue our consideration of equations with gradient nonlinearities. In this paper, we consider initial-boundary value problem in a bounded domain with smooth boundary for non-local in time equation with gradient nonlinearity and prove the local solvability of the strong generalized sense, in addition, we obtain sufficient conditions for the destruction of a finite time and sufficient conditions for global in time solubility.

Keywords

nonlocal equation with gradient nonlinearity, Sobolev type equations, the destruction of the solutions.

References

1. Sviridyuk G.A. On the General Theory of Operator Semigroups. Russ. Math. Surv., 1994, vol. 49, no. 4, pp. 45 - 74.
2. Sviridyuk G.A. One Problem for the Generalized Boussinesq Filtration Equation [Odna zadacha dlya obobshchennogo fil'tratsionnogo uravneniya Bussineska]. Russian Mathematics, 1989, no. 2, pp. 55 - 61.
3. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev type. Russian Acad. Sci. Izv. Math., 1994, vol. 42, no. 3, pp. 601 - 614.
4. Sviridyuk G.A. Phase Spaces of Semilinear Equations of Sobolev Type with Relatively Strongly Sectorial Operator [Fazovye prostranstva polulineynykh uravneniy tipa Soboleva s otnositel'no sil'no sektorial'nym operatorom]. Algebra i analiz, 1994, vol. 6, issue 5, pp. 252 - 272.
5. Korpusov M.O. Blow-up Solutions of the Equation with Gradient Nonlinearity [O razrushenii resheniya uravneniya s gradientnoy nelineynost'yu]. Differential Equations, in print.
6. Bonch-Bruevich V.L., Kalashnikov S.G. Fizika poluprovodnikov [Physics of Semiconductors]. Moscow, Nauka, 1990. 672 p.
7. Bonch-Bruevich V.L., Zvyagin I.P., Mironov A.G. Domennaya elektricheskaya neustoychivost' v poluprovodnikakh [Blast Electric Instability in Semiconductors]. Moscow, Nauka, 1972. 417 p.
8. Landau L.D., Lifshits E.M. Elektrodinamika sploshnykh sred. Teoreticheskaya fizika. T. 8 [Electrodynamics of Continuous Media. Theoretical Physics. Vol. 8]. Moscow, Nauka, 1992. 664 p.
9. Bass F.G., Bochkov V.S., Gurevich Yu.S. Elektrony i fonony v ogranichennykh poluprovodnikakh [Electrons and Phonons in a Limited Semiconductors]. Moscow, Nauka, 1984. 288 p.
10. Lions Zh.-L. Nekotorye metody resheniya nelineynykh kraevykh zadach [Some Methods for Solving Nonlinear Boundary Value Problems]. Moscow, Mir, 1972. 588 p.
11. Demidovich V.P. Lektsii po matematicheskoy teorii ustoychivosti [Lectures on the Mathematical Theory of Stability]. Moscow, Nauka, 1967. 472 p.
12. Samarskiy A.A., Galaktionov V.A., Kurdyumov S.P., Mikhaylov A.P. Rezhimy s obostreniem v zadachakh dlya kvazilineynykh parabolicheskikh uravneniy [Regimes with Peaking in Problems for Quasilinear Parabolic Equations]. Moscow, Nauka, 1987. 480 p.
13. Vatson G.N. Teoriya Besselevykh funktsiy. Chast' I [The theory of Bessel functions. Part I] Moscow, Izd-vo Inostr. lit., 1949. 800 p.