This paper is concerned with the qualitative analysis of a class of second-order nonlinear evolution inclusions. The feature of this paper is the presence of non-monotone and noncoercive viscosity term. The novelty of this paper is that develops a kind of parabolic regularization of the inclusion, initially introduced by Jacques-Louis Lions in the context of semilinear hyperbolic equations. In such a way, using the nonsmooth Clarke theory in combination with a priori estimates that allow to pass to the limit, we obtain a sufficient condition for the existence of solutions. The main abstract result of this paper is illustrated with a hyperbolic boundary value problem with forcing term. The methods introduced in this paper can be extended to other classes of nonlinear evolutionary inequality problems. This paper extends in the framework of evolution inclusions some ideas developed in our monograph V. Radulescu, D. Repovš, Partial differential equations with variable exponents: Variational methods and qualitative analysis, Francis & Taylor, New York, 2015 [COBISS.SI-ID 17325401] in the framework of problems with variable exponent.
The content of this paper is at the interplay between applied nonlinear analysis and optimal control theory. We are concerned with a nonlinear optimal control problem governed by a nonlinear evolution inclusion and depending on a parameter. We are mainly interested in the sensitivity analysis of this class of problems. For this purpose, we first examine the dynamics of the problem and establish the non-emptiness of the solution set. Next, we produce continuous selections of the solution multifunction in relationship with the initial condition. The feature of this paper is that the main results are obtained. These results are in a very general abstract setting. These results are of independent interest as results concerning the qualitative and sensitivity analysis of evolution inclusions. The results developed in this paper allow to establish the Hadamard well-posedness (continuity of the value function), and to argue the continuity properties of the optimal multifunction. The final part of this paper includes an application to the theory of nonlinear parabolic distributed parameter systems.
This paper is concerned with a refined mathematical analysis of solutions of a class of nonlinear stationary equations on the Sierpinski gasket and other self-similar fractal domains. The main difficulty stems from the fact that the standard differential operators (starting with the Laplace operator) have a different structure as in the standard setting and their definition strongly depends on the fractal on which they are defined. This paper introduces variational and topological methods and we establish sufficient conditions for the existence of nontrivial solutions in an appropriate function space. The arguments combine the force of the mountain pass theorem and other critical point theorems with related topological methods (deformation or Morse theory). The content of this talk complements in this case of non-standard domains some results included in our monograph G. Molica Bisci, V. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, Cambridge University Press, Cambridge 2016 [COBISS.SI-ID 17642841]. Moreover, the methods introduced in this paper can be extended to nonlinear problems on other fractal sets (Sierpinski carpet, Julia set, Mandelbrot fractal, etc.).