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Nonlinear evolution equations that are non-local in space and time. (English) Zbl 1372.45008

Summary: We deal with a nonlocal nonlinear evolution problem of the form \[ \mathop{\iint}\limits_{\mathbb{R}^n \times \mathbb{R}} J(x - y, t - s) | \overline{v}(y, s) - v(x, t) |^{p - 2}(\overline{v}(y, s) - v(x, t)) d y d s = 0 \] for \((x, t) \in \mathbb{R}^n \times [0, \infty)\). Here \(p \geq 2\), \(J : \mathbb{R}^{n + 1} \rightarrow \mathbb{R}\) is a nonnegative kernel, compactly supported inside the set \(\{(x, t) \in \mathbb{R}^{n + 1} : t \geq 0 \}\) with \(\iint_{\mathbb{R}^n \times \mathbb{R}} J(x, t) d x d t = 1\) and \(\overline{v}\) stands for an extension of a given initial value \(f\), that is, \[ \overline{v}(x, t) = \begin{cases} v(x, t) & t \geq 0, \\ f(x, t) & t < 0. \end{cases} \] For this problem we prove existence and uniqueness of a solution. In addition, we show that the solutions approximate viscosity solutions to the local nonlinear partial differential equation \(\|\nabla u \|^{p - 2} u_t = \Delta_p u\) when the kernel is rescaled in a suitable way.

MSC:

45G10 Other nonlinear integral equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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