| Abstract: |
| In this talk, we evaluate lifespan of the solution of the equation obtained by discretising a semilinear wave equation with a power-type nonlinear term. The discrete equation was first proposed by Matsuya(2013) and has the following form.
$\begin{align*}
u_n^{t+1}+u_n^{t-1} &= \dfrac{4 v_n^t}{2-\delta^2v_n^t|v_n^t|^{p-2}}\quad (n \in \mathbb{Z}^d,\, t \in \mathbb{Z}_{\ge 0}) \
v_n^t &:= \frac{1}{2 d}\sum_{i=1}^d\left(u^t_{n+e_i}+u^t_{n-e_i}\right)
\end{align*}$
In continuous limit, this discrete equation turns to the semiliniar wave equation:
$\begin{align*}
u_{tt}=\Delta u +|u|^p
\end{align*}$
This semiinear wave equation is known to explode if the exponent $p$ appearing in the nonlinear term is smaller than a certain value when the initial conditions are sufficiently small, and the discrete equation has been proved to have similar behaviour to the original wave equation. We show that the discrete equation also has similar lifesapn to that of the semilinear wave equation.
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Reference: Keisuke Matsuya (2013), A blow-up theorem for a discrete semilinear wave equation, Journal of Difference Equations and Applications, 19:3, 457-465 |
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