Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces

Abstract

Let V and V * be a real reflexive Banach space and its dual space, respectively. This paper is devoted to the abstract Cauchy problem for doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations of the form:the subdifferential operators of proper, lower semicontinuous and convex functions ψ t , ϕ : V → (−∞, +∞], respectively, for each t ∈ [0, T ], and f : (0, T ) → V * and u 0 ∈ V are given data. Moreover, let B be a (possibly) multi-valued operator from (0, T )× V into V * . We present sufficient conditions for the local (in time) existence of strong solutions to the Cauchy problem as well as for the global existence. Our framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e., B ≡ 0) might not be uniquely solved in a doubly nonlinear setting. Our proof relies on a couple of approximations for the equation and a fixed point argument with a multi-valued mapping. Moreover, the preceding abstract theory is applied to doubly nonlinear parabolic equations.

Optimal Control of Doubly Nonlinear Evolution Equations Governed by Subdifferentials Without Uniqueness of Solutions

“…In this section, we give the sketch of the proof of Proposition 2 by arguments similar to Akagi [1] and Aso et al [3]. Moreover, we prove Theorem 1.…”

“…By a slight modification of [1,3], we can prove the following existence result for problem (P;f ). We give a sketch of its proof in Section 3.…”

“…Also, Kadoya-Kenmochi [12] studied the optimal sharp design of elliptic-parabolic equations. On the other hand, Akagi [1], Arai [2], Aso et al [3,4], Colli [8], Colli-Visintin [9], Senba [21] investigated the following type of doubly nonlinear evolution equations (cf. (1.1)):…”

Porous medium equation with a blow-up nonlinearity and a non-decreasing constraint

“…A(u t ) + B(u) = 0 and ∂ t A(u) + B(u) = 0 with two nonlinear operators A and B. The former one appears in the study of generalized Ginzburg-Landau equations (see [23] and also [2] with references therein), unidirectional heat flow (see [3]) and so on (see also [4,7,18,19,31,32,37,[39][40][41]43,44]). The latter one represents nonlinear diffusion equations, e.g., porous medium/fast diffusion equations and Stefan problem.…”

Periodic solutions for doubly nonlinear evolution equations

“…Later on the problem has been considered also by Colli & Visintin [20] in Hilbert spaces and Colli [18] in Banach spaces. Besides existence, the Cauchy problem has also been considered from the point of view of structural stability [2], perturbations and long-time behavior [3,4,8,34,35,36], and variational characterization of solutions [6,5,38,32,40]. The interest in the study of periodic solutions is in particular to be considered as a further step toward the comprehension of long-time dynamics and bifurcation phenomena.…”