One may have an equation (with boundary conditions omitted below) $$u_t - Au = f$$ $$u(0)=u_0$$ which has a weak solution $u \in L^2(0,T;V) \cap C([0,T];H)$ in the sense that $$-\int_0^T \int_\Omega u(t)\phi_t(t) + \int_0^T\int_{\Omega} A^{\frac 12} u(t) A^{\frac 12} \phi(t) = \int_0^T \int_\Omega f(t)\phi(t) + \int_\Omega u_0\phi(0)$$ for all test functions $\phi$ vanishing at $t=T$. Since we do not know if $u_t$ exists or is a function, the concept of Steklov averages $$v_h(t) = \frac 1h\int_t^{t+h}v(s)\;\mathrm{d}s$$ is useful, since the weak formulation above can be rewritten as something like $$\int_0^T \int_\Omega \partial_t u_h(t)\phi(t) + \int_0^T\int_{\Omega} A^{\frac 12} u_h(t) A^{\frac 12}\phi(t) = \int_0^T \int_\Omega f(t)\phi(t) + \int_\Omega u_0\phi(0)$$ and then we can test and obtain some bounds depending on the application.

This doesn't work when the operator $A=A(t)$ is time-dependent, since the integral in the Steklov operator cannot be commuted with the $A(t).$

Does anyone know what the alternatives are to this? Maybe there is a modification?

Motivation: I am trying to prove that the time derivative of a solution to a PDE is a function in $L^p$ space.


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