On Equations of the Form $\Delta u-\frac{\partial u}{\partial t} =f\left(x,t,u,Du\right)$
R.A. Amanov, A.I. Ismailov
Abstract. The paper describes an interpolation method for obtaining a priori estimates for strong solutions of semilinear parabolic equations with unbounded singularities on the right-hand side, provided that there is a first a priori estimate in the space of summable functions.
KeyWords and Phrases: a priori estimate, semilinear, maximum principle.
2000 Mathematics Subject Classiffcations: 35B50, 35K61.
References.
[1] S.N. Bernstein, On the equations of the calculus of variations, Collected
Works, Moscow, ed. USSR Academy of Sciences, 1960, 191-241.
[2] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Equations of
Elliptic Type, Moscow, Nauka, 1973.
[3] H. Amann, M.G. Grandall, On some existence theorems for semilinear el-
liptic equations, Ind. Univ. Math. J., 27(5), 1978, 779-790.
[4] J.L. Kazdan, R.J. Kramer, Invariant criteria for existence of solutions to
second-order quasilinear elliptic equations, Comm. Pure and Appl. Math.,
31(5), 1978, 619-645.
[5] S.I. Pokhozhaev, On equations of the form u = f (x; u;Du), Mat. Sb.,
113(155), 2(10), 1980, 324-338.
[6] G.G. Laptev, On an interpolation method for obtaining a priori estimates
for strong solutions of second-order semilinear parabolic systems, Trudy Mat.
Inst. them. V.A. Steklova, 227, 1999, 180-191.
[7] N.V. Krylov, Nonlinear elliptic and parabolic equations of the second order,
Moscow, Nauka, 1985.
[8] O.V. Besov, V.P. Ilin, S.M. Nikolski, Integral Representation of Functions
and Embedding Theorem, Moscow, Nauka, 1996.
[9] S.L. Sobolev, Some applications of functional analysis in mathematical
physics, Novosibirsk, SO ANSSSR, 1962.
|