Strong Formulation

We seek functions $u$ and $z$ satisfying the following system of equations:

\[\begin{align}\label{pde:mdl} \begin{aligned} & u^{\prime\prime}(x,t) - \alpha(t)\Delta u(x,t) + f\big(u(x,t)\big) = f_1(x,t), \quad(x,t)\in\Omega\times(0,+\infty), \\[5pt] & q_1 z^{\prime\prime}(x,t) + q_2 z^\prime(x,t) + q_3 z(x,t) + q_4 u^\prime(x,t) = f_2(x,t), \quad(x,t)\in\Gamma_1\times(0,+\infty), \\[5pt] & \frac{\partial u}{\partial\nu}(x,t) = z^\prime(x,t) - g\big(x,u^\prime(x,t)\big), \quad(x,t)\in\Gamma_1\times(0,+\infty), \\[5pt] & u(x,t) = 0,\quad (x,t)\in\Gamma_0\times(0,+\infty), \end{aligned} \end{align}\]

with initial conditions

\[\begin{align}\label{pde:mdl:initial_condition} \begin{aligned} & u(x,0) = u_0(x),\quad u^\prime(x,0)=v_0(x),\quad x\in\Omega, \\ & z(x,0) = z_0(x),\quad z^\prime(x,0) = r_0(x) \equiv \frac{\partial u_0}{\partial\nu}(x) + g\big(x,v_0(x)\big),\quad x\in\Gamma_1, \end{aligned} \end{align}\]

where $\Omega$ is a bounded open subset of $\mathbb{R}^n$, $n\geq 2$, with smooth boundary $\Gamma=\Gamma_0\cup\Gamma_1$ and disjoint $\Gamma_0$, $\Gamma_1$.

Existence and uniqueness results for particular cases of \eqref{pde:mdl}–\eqref{pde:mdl:initial_condition} can be found in:

Alcântara et al. (2025). Numerical analysis for nonlinear wave equations with boundary conditions: Dirichlet, Acoustics and Impenetrability. Applied Mathematics and Computation. https://doi.org/10.1016/j.amc.2024.129009

Weak Formulation

We seek functions $u(t)\in H_{\Gamma_0}^1(\Omega)$ and $z(t)\in L^2(\Gamma_1)$ such that

\[\begin{align} \label{pde:variational_form} \begin{aligned} & \big(\varphi,u^{\prime\prime}(t)\big) + \alpha(t)\Big[ \big(\nabla\varphi,\nabla u(t)\big) - \big(\varphi,z^\prime(t))_{\Gamma_1} + \big(\varphi,g\big(u^\prime(t)\big)\big)_{\Gamma_1}\Big] + \big(\varphi,f\big(u(t)\big)\big) = \big(\varphi,f_1(t)\big), \quad\forall\varphi\in H_{\Gamma_0}^1(\Omega), \\ & \big(\phi,q_1z^{\prime\prime}(t) + q_2z^\prime(t) + q_3z(t) + q_4u^\prime(t)\big)_{\Gamma_1} = \big(\phi,f_2(t)\big)_{\Gamma_1}, \quad\forall\phi\in L^2(\Gamma_1), \end{aligned} \end{align}\]

with $u(0)=u_0$, $u^\prime(0)=v_0$, $z(0)=z_0$, and $z^\prime(0) = r_0 \equiv \frac{\partial u_0}{\partial\nu} + g(v_0)$.

By introducing the auxiliary variables $v(t)=u^\prime(t)$ and $r(t)=z^\prime(t)$, we obtain the equivalent first-order system: find functions $u(t),v(t)\in H_{\Gamma_0}^1(\Omega)$ and $z(t),r(t)\in L^2(\Gamma_1)$ such that

\[\begin{align} \label{pde:variational_form_opt2} \begin{aligned} & \big(\varphi,v^{\prime}(t)\big) + \alpha(t)\Big[ \big(\nabla\varphi,\nabla u(t)\big) - \big(\varphi,r(t)\big)_{\Gamma_1} + \big(\varphi,g\big(v(t)\big)\big)_{\Gamma_1}\Big] + \big(\varphi,f\big(u(t)\big)\big) = \big(\varphi,f_1(t)\big), \quad\forall\varphi\in H_{\Gamma_0}^1(\Omega), \\ & \big(\phi,q_1r^{\prime}(t) + q_2r(t) + q_3z(t) + q_4v(t)\big)_{\Gamma_1} = \big(\phi,f_2(t)\big)_{\Gamma_1}, \quad\forall\phi\in L^2(\Gamma_1), \\ & u^\prime(t)=v(t),\quad z^\prime(t)=r(t), \end{aligned} \end{align}\]

with initial conditions $u(0)=u_0$, $v(0)=v_0$, $z(0)=z_0$, and $r(0) = r_0 \equiv \frac{\partial u_0}{\partial\nu} + g(v_0)$.