## Osaka Journal of Mathematics

### STRONG-VISCOSITY SOLUTIONS: CLASSICAL AND PATH-DEPENDENT PDEs

#### Abstract

The aim of the present work is the introduction of a viscosity type solution, called $strong$-$viscosity$ $solution$ emphasizing also a similarity with the existing notion of $strong$ $solution$ in the literature. It has the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.

#### Article information

Source
Osaka J. Math., Volume 56, Number 2 (2019), 323-373.

Dates
First available in Project Euclid: 3 April 2019

https://projecteuclid.org/euclid.ojm/1554278428

Mathematical Reviews number (MathSciNet)
MR3934979

Zentralblatt MATH identifier
07080088

#### Citation

Cosso, Andrea; Russo, Francesco. STRONG-VISCOSITY SOLUTIONS: CLASSICAL AND PATH-DEPENDENT PDEs. Osaka J. Math. 56 (2019), no. 2, 323--373. https://projecteuclid.org/euclid.ojm/1554278428

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