Existence of weak solutions for a nonlinear parabolic equations by Topological degree

Öz

We study the nonlinear parabolic initial boundary value problem associated to the equation ut − diva(x, t, u, grad u) = f(x, t), where the terme − diva(x, t, u, grad u) is a Leray-Lions operator, The right-hand side f is assumed to belong to L^q(Q). We prove the existence of a weak solution for this problem by using the Topological degree theory for operators of the form L + S, where L is a linear densely defined maximal monotone map and S is a bounded demicontinuous map of class (S+) with respect to the domain of L.

Anahtar Kelimeler

• Nonlinear parabolic equations
• Topological degree
• Weak solution
• map of class (S+)

Kaynakça

1. [1] Asfaw T. M., A degree theory for compact perturbations of monotone type operators and application to nonlinear parabolic problem. Abstract and Appl. Anal.(2017):13 pages.
2. [2] Berkovits J. and Mustonen V., Topological degree for perturbations of linear maximal monotone mappings and applications to a class of parabolic problems. Rend. Mat. Appl. 12 no 3 (1992), 597-621.
3. [3] Boccardo B., Dall’Aglio A., Gallou¨ot T. and Orsina L., Existence and regularity results for some nonlinear parabolic equations. Adv. Math. Sci. Appl. 9 no 2 (1999), 1017-1031.
4. [4] Browder F. E., Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. Contributions to Nonlinear Analysis (E. Zarantonello, ed.). Academic Press, New York, 1971.
5. [5] Lions J. L., Quelques m´ethodes de resolution des problmes aux limites non-lineaires. Dunod, Paris, 1969.
6. [6] Zeidler E., Nonlinear Functional Analysis and its Applications. Springer-Verlag, New York, 1990.