For generalized solutions of the Cauchy problem \[ {\partial^2v\over\partial t^2}+ Av= f(t,x),\quad t\in\mathbb{R}^1,\quad x\in\mathbb{R}^2, \]
\[ v(0,x)= v_0(x),\quad {\partial v(0,x)\over\partial t}= v_1(x),\quad x\in\mathbb{R}^2 \] the following energetical inequality (in \(L_p(\mathbb{R}^2)\), for \(p>1\)) \[ \begin{split} \Biggl\|{\partial v(t,x)\over\partial t}\Biggr\|_{L_p(\mathbb{R}^2)}+ \| v(t,x)\|_{W^1_p(\mathbb{R}^2)}\leq \Biggl(\mu{p^2\over p-1}\Biggr)^3 e^{2| t|}\Biggl[\| v_0(x)\|_{W^1_p(\mathbb{R}^2)}+ \| v_1(x)\|_{L_p(\mathbb{R}^2)}+\\ \text{sign }t \int^t_0 \exp\{2| t-s|- 2| t|\}\| f(s,x)\|_{L_p(\mathbb{R}^2)}ds\Biggr]\end{split} \] has been proved. Here, \(A\) is a pseudodifferential elliptic operator of the second order.
This result is a generalization of the energetical inequality in \(L_p(\mathbb{R}^1)\) for \(p\geq 1\) for the wave equation.