Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals

We investigate the existence of strong solutions to a general class of doubly multivalued and nonlinear evolution equations of second-order. The multivalued operators are generated by the subdifferential of nonsmooth potentials that live in different spaces, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>U</m:mi> </m:math> U and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V , where in general <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>U</m:mi> <m:mspace width="0.33em"/> <m:mo>⊈</m:mo> <m:mspace width="0.33em"/> <m:mi>V</m:mi> </m:math> U{0.33em} {0.33em}V and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> <m:mspace width="0.33em"/> <m:mo>⊈</m:mo> <m:mspace width="0.33em"/> <m:mi>U</m:mi> </m:math> V{0.33em} {0.33em}U . The proof is based on the regularization of the dissipation potential using the generalized Moreau-Yosida regularization and a semi-implicit time-discretization scheme, which demonstrates the existence of strong solutions to the regularized problem. The existence of solutions to the original problem is then shown by letting the regularization parameter converge to zero. Furthermore, we establish an energy-dissipation inequality for the solution. We conclude with applications of this abstract theory.