Speaker
Description
\textbf{Abstract.}
We establish well-posedness and strong stability for a degenerate wave equation with tempered fractional damping, complemented by an fPINN numerical approximation.
\textbf{Keywords:} Degenerate wave equation; tempered Caputo damping; $C_0$-semigroup; strong stability; fPINN.
\section{Model and Main Results}
Let $\Omega=(0,1)$ and consider
[
u_{tt}-(x^m u_x)_x+\beta u+
\gamma D_t^{\alpha,\eta}u_t=0,
\qquad
u(0,t)=u(1,t)=0,
]
where $0<\alpha<1$, $m>0$, $\beta,\gamma>0$, and
[
D_t^{\alpha,\eta}w(t)=
\frac{1}{\Gamma(1-\alpha)}
\int_0^t(t-s)^{-\alpha}e^{-\eta(t-s)}w'(s)\,ds.
]
Define
[
V_m={u\in L^2(0,1):x^{m/2}u_x\in L^2(0,1),
u(0)=u(1)=0},
]
and
[
Au=-(x^m u_x)x+\beta u,
\qquad
a(u,u)\geq\beta|u|^2.
]
Using the memory variable
[
z_t+(\xi^2+\eta)z=v,\qquad v=u_t,
]
the problem is formulated on
[
\mathcal H=V_m\times L^2(0,1)\times\mathcal M
]
as $U_t=\mathcal AU$. Moreover,
[
\operatorname{Re}\langle\mathcal AU,U\rangle{\mathcal H}
=-\gamma\int_{\mathbb R}(\xi^2+\eta)M(\xi)
|z(\xi)|^2d\xi\leq0.
]
Surjectivity of $\lambda I-\mathcal A$ and the Lumer--Phillips theorem imply that $\mathcal A$ generates a contraction $C_0$-semigroup. Spectral analysis and the Arendt--Batty--Lyubich--Vũ theorem yield
[
\lim_{t\to\infty}|S(t)U_0|_{\mathcal H}=0.
]
\section{fPINN Approximation}
We set
[
u_\theta=x(1-x)\mathcal N_\theta(x,t),
]
with residual
[
\mathcal R_\theta=(u_\theta){tt}
-\partial_x(x^m(u\theta)x)+\beta u\theta
+\gamma D_t^{\alpha,\eta}(u_\theta)t.
]
The network minimizes
[
\mathcal L=\mathcal L{\rm PDE}
+\lambda_u\mathcal L_{u_0}
+\lambda_v\mathcal L_{v_0}.
]
Numerically, we take
[
m=1,\quad\alpha=0.5,\quad\eta=\gamma=1,\quad T=2,
\qquad
u_0(x)=\sin(\pi x),\quad u_1(x)=0.
]
The fPINN approximation complements the theoretical stability analysis while preserving the nonlocal fractional-memory effect.
\section*{Conclusion}
Well-posedness and strong stability are established via semigroup and spectral analysis, while fPINN simulations validate the dynamics and fractional-memory effects.
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\bibitem{Deng2022}
S. Deng and M. J. Alves,
\textit{Well-posedness and stability of wave equations with fractional damping},
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