Strongly continuous semigroup

Question

Given the following PDE $$\left\{\begin{array}{lll}u'(t)&=&A(u(t)) + f(t)\ \mathrm{for\ all}\ t \geq 0\ ,\\ u(0)&=&u_0\\\end{array}\right.$$

with $D(A) = H^2(0,1) \cap H_0^1((0,1))$

Now assume that $(T(t))_{t \ge 0 }$ is the strongly continuous semigroup generated by $A$, $u_0$ should be a $L^2$ function and $f$ a $L^2$ function. Now my question is:

The solution to this PDE is given as

$$u(t) := T(t)u_0 + \int_0^t T(t-s)f(s)$$

Is this expression then continuous in time?

My idea is that the first term is continuous in time, as $T$ is strongly-continuous. Moreover, the integral is continuous and therefore I think it is very likely that also the second term is continuous in time, but somehow we need to justify that also the composition of the integral and the integrand is continuous. How can I show this?

Answer 1

Your intuition works perfectly, let's write down the details.

We need to prove that $u(t + h) - u(t) \to 0$ as $h \to 0$.

Let's first assume that $h \to 0^+$: \begin{align} u(t + h) - u (t) = &\ \big(T(t + h) - T(t)\big)u_0 + \int_0^{t + h}T(t + h - s)f(s) - \int_0^tT(t - s)f(s) \\ = &\ T(t)\big(T(h)u_0 - u_0\big) + \int_0^t\big(T(t + h - s) - T(t - s)\big)f(s) + \int_t^{t + h}T(t + h - s)f(s) \\ = &\ T(t)\big(T(h)u_0 - u_0\big) + (T(h) - I)\Big(\int_0^tT(t - s)f(s)\Big) + \int_t^{t + h}T(t + h - s)f(s). \end{align}

The first two terms tend to $0$ as $h \to 0$ since $T$ is strongly continuous (hence it approaches the identity operator as its argument goes to $0$). To deal with the last integral notice that $$\Big\|\int_t^{t + h}T(t + h - s)f(s)\Big\|\le M\int_t^{t + h}e^{\omega (t + h - s)}|f(s)| \to 0\ \text{as}\ h \to 0.$$

This concludes the proof for $t = 0$. If $t \neq 0$ we need to study the limit from the left. This case can be treated similarly, indeed let's consider $h \to 0^+$ and expand $u(t - h) - u(t)$ considering values of $h$ for which $t - h \ge 0$: \begin{align} u(t - h) - u(t) = &\ \big(T(t - h) - T(t)\big)u_0 + \int_0^{t - h}T(t - h - s)f(s) - \int_0^tT(t - s)f(s) \\ = &\ T(t - h)\big(u_0 - T(h)u_0\big) + \int_0^{t - h}\big(T(t - h -s) - T(t - s)f(s)\big) + \int_{t - h}^tT(t -s)f(s) \\ \to &\ 0. \end{align}