Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

Abstract

The (third order in time) JMGT equation [Jordan (J Acoust Soc Am 124(4):2491–2491, 2008) and Cattaneo (C Sulla conduzione del calore Atti Sem Mat Fis Univ Modena 3:83–101, 1948)] is a nonlinear (quasi-linear) partial differential equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second-order in time equation referred to as Westervelt equation. Replacing Fourier’s law by Maxwell–Cattaneo’s law gives rise to the third-order in time derivative scaled by a small parameter $\tau >0$, the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper, we provide an asymptotic analysis of the third-order model when $\tau \rightarrow 0$. It is shown that the corresponding solutions converge in a strong topology of the phase space to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher-order regularity. This addresses an open question raised in [20], where a related JMGT equation has been studied and weak star convergence of the solutions when $\tau \rightarrow 0$ has been established. Thus, our main contribution is showing strong convergence on infinite time horizon, along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the “smallness” of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.

A Proof of Theorem 1.4

As a starting point, we are going to use (see [1, 18, 19]) that given $f \in L^1(0,T; \mathcal {D}(A^{1/2}))$ and $\alpha > 0$, the linear problem

$$\begin{aligned} \tau u_{ttt} + \alpha u_{tt} + c^2A u + bA u_t = f, \end{aligned}$$

(A.1)

is well posed—in the variable $U = (u,u_t,u_{tt})$—and exponentially stable (with rates independent of $\tau$ for $\tau$ small) for initial data in $\mathbb {H}_i$ ($i = 0,1,2$). This is to say that, denoting by $t \mapsto S(t)$ the semigroup generated by the evolution (A.1), there exist constants $\omega _i, M_i >0$, ($i = 0,1,2$) such that

$$\begin{aligned} \Vert U(t)\Vert _{\mathbb {H}_i^\tau } = \Vert S(t)U_0\Vert _{\mathbb {H}_i^\tau } \leqslant M_ie^{-\omega _i t} \Vert U_0\Vert _{\mathbb {H}_i^\tau }. \end{aligned}$$

(A.2)

Define X as the set

$$\begin{aligned} X = \left\{ W = (w,w_t,w_{tt})^\top \in C(0,T; \mathbb {H}_2); \sup _{t \in [0,T]} \Vert W(t)\Vert _{\mathbb {H}_2^\tau }< \infty \ \text{ and } \ \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_0^\tau } < \eta \right\} \end{aligned}$$

($\eta > 0$ will be taken to be sufficiently small later) and equip it with the norm

$$\begin{aligned} \Vert W\Vert _{X}^2 := \sup _{t \in [0,T]} \Vert W(t)\Vert _{\mathbb {H}_2^\tau }^2. \end{aligned}$$

Recalling the interpolation inequalities

$$\begin{aligned} \Vert g\Vert _{L^\infty } \leqslant C\Vert g\Vert _{\mathcal {D}(A^{1/2})}^{1/2}\Vert g\Vert _{\mathcal {D}(A)}^{1/2}, \ g \in \mathcal {D}(A) \end{aligned}$$

(A.3)

and

$$\begin{aligned} \Vert g\Vert _{L^4} \leqslant C\Vert g\Vert _2^{1/4}\Vert g\Vert _{\mathcal {D}(A^{1/2})}^{3/4} \ \ g \in \mathcal {D}(A^{1/2}) \end{aligned}$$

(A.4)

we observe that if $W = (w,w_t,w_{tt}) \in X$, it follows that ${w},{w}_t \in \mathcal {D}(A) \hookrightarrow L^\infty (\Omega )$ and $w_{tt} \in \mathcal {D}(A^{1/2}) \hookrightarrow L^4(\Omega )$ ($n = 2,3,4$), for each $t \in [0,T]$, and therefore $f({w}) := 2k({w}_t^2 + {w}{w}_{tt}) \in C(0,T;\mathcal {D}(A^{1/2})).$ This means that for each $W \in X$, f(w) qualifies to be the right-hand side of (A.1), and therefore, it makes well defined the application $\Upsilon$ that associates each $W \in X$ to the solution $(u,u_t,u_{tt})^\top = U := \Upsilon (W) \in C(0,T; \mathbb {H}_2)$ for (A.1) with initial condition $U_0 = (u(0),u_t(0),u_{tt}(0)) ^\top \in \mathbb {H}_2$. Moreover, the solution U is represented by the variation of parameters formula, i.e., for each $t \in [0,T],$

$$\begin{aligned} U(t) = \Upsilon (W)(t) = S(t)U_0 + \int _0^t S(t-\sigma )\underbrace{(0,0,\tau ^{-1}f(w(t)))^\top }_{:=F_\tau (W)(t)} d\sigma . \end{aligned}$$

(A.5)

In addition, $\Upsilon$ maps X into itself. In fact, for each $t \in [0,T]$, uniform (in $\tau$) exponential stability implies that

$$\begin{aligned} \Vert \Upsilon (W)(t)\Vert _{\mathbb {H}_2^\tau }&\leqslant \Vert S(t)U_0\Vert _{\mathbb {H}_2^\tau } + \left\| \int _0^tS(t-\sigma )F_\tau (W)(\sigma )d\sigma \right\| _{\mathbb {H}_2^\tau } \nonumber \\&\leqslant M_2\Vert U_0\Vert _{\mathbb {H}_2^\tau } + \int _0^t M_2e^{-\omega _2 (t-\sigma )} \Vert F_\tau (W)(\sigma )\Vert _{\mathbb {H}_2^\tau }d\sigma \nonumber \\&\leqslant M_2\left( \Vert U_0\Vert _{\mathbb {H}_2^\tau } + \dfrac{C_{\omega }}{\tau }\sup _{t \in [0,T]} \Vert f(w)(t)\Vert _{\mathcal {D}(A^{1/2})}\right) . \end{aligned}$$

(A.6)

and again for each $t \in [0,T]$—mostly omitted on the computations below—we estimate

$$\begin{aligned}&(2k)^{-1}\Vert f({w})\Vert _{\mathcal {D}(A^{1/2})}\nonumber \\&\quad \sim \Vert \nabla ({w}_t^2 + {w} {w}_{tt})\Vert _2 \nonumber \\&\quad = \Vert 2{w}_t \nabla {w}_t + {w} \nabla {w}_{tt} + {w}_{tt} \nabla {w}\Vert _2 \nonumber \\&\quad \leqslant 2\Vert {w}_t\Vert _{L^\infty } \Vert \nabla {w}_t\Vert _2 + \Vert {w}\Vert _{L^\infty }\Vert \nabla {w}_{tt}\Vert _2 + \Vert {w}_{tt}\Vert _{L^4}\Vert \nabla {w}\Vert _{L^4} \nonumber \\&\quad \leqslant C \left[ \Vert {w}_t\Vert _{\mathcal {D}(A^{1/2})}^{1/2}\Vert {w}_t\Vert _{\mathcal {D}(A)}^{1/2}\Vert \nabla {w}_t\Vert _2 + \Vert {w}\Vert _{\mathcal {D}(A^{1/2})}^{1/2}\Vert {w}\Vert _{\mathcal {D}(A)}^{1/2}\Vert \nabla {w}_{tt}\Vert _2\right] \nonumber \\&\qquad + C\left[ \Vert {w}_{tt}\Vert _2^{1/4}\Vert {w}_{tt}\Vert _{\mathcal {D}(A^{1/2})}^{3/4}\Vert \nabla {w}\Vert _2^{1/4}\Vert \nabla {w}\Vert _{\mathcal {D}(A^{1/2})}^{3/4}\right] \nonumber \\&\quad \leqslant C\Vert W(t)\Vert _{\mathbb {H}_0^\tau }^{1/2}\Vert W(t)\Vert _{\mathbb {H}_2^\tau }^{3/2} \leqslant \left[ \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/2}\eta ^{1/2}, \end{aligned}$$

(A.7)

and then, back in (A.6) we conclude that

$$\begin{aligned} \sup _{t \in [0,T]}\Vert \Upsilon (W)(t)\Vert _{\mathbb {H}_2^\tau } \leqslant M_2\left( \Vert U_0\Vert _{\mathbb {H}_2^\tau } + \dfrac{2k C_{\omega }}{\tau }\left[ \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/2}\eta ^{1/2}\right) < \infty .\nonumber \\ \end{aligned}$$

(A.8)

Similarly, for $\mathbb {H}_0^\tau$, for each $t \in [0,T]$ we have have

$$\begin{aligned} \Vert \Upsilon (W)(t)\Vert _{\mathbb {H}_0^\tau } \leqslant M_1\left( \Vert U_0\Vert _{\mathbb {H}_0^\tau } + \dfrac{C_{\omega }}{\tau }\sup _{t \in [0,T]}\Vert f(w)(t)\Vert _2\right) . \end{aligned}$$

(A.9)

Moreover, for each $t \in [0,T]$—again mostly omitted—it holds that

$$\begin{aligned}&(2k)^{-1}\Vert f(w)(t)\Vert _2 \\&\quad = \Vert w_t^2 + uu_{tt}\Vert _2 \\&\quad \leqslant \Vert w_t\Vert _{L^4}^2 + \Vert u\Vert _{L^\infty }\Vert u_{tt}\Vert _2 \\&\quad \leqslant C\left[ \Vert A^{1/2}w_t\Vert _2^2 + \Vert A^{1/2}u\Vert _2^{1/2}\Vert Au\Vert _2^{1/2}\Vert u_{tt}\Vert _2\right] \\&\quad \leqslant C\left\{ \left[ \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_0^\tau }\right] ^2 + \left[ \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{3/2}\left[ \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{1/2}\right\} \\&\quad \leqslant C\left\{ \eta ^2 + \eta ^{3/2}\left[ \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{1/2}\right\} , \end{aligned}$$

then, taking $\eta = \eta (\tau )$ small enough so that

$$\begin{aligned} \dfrac{2kC}{\tau }\left[ \eta ^2 + \eta ^{3/2}\left( \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_2^\tau }\right) ^{1/2}\right] < \dfrac{\eta }{2M1} \end{aligned}$$

and, as a consequence, $\rho = \rho (\tau )$ small enough such that $\rho < \dfrac{\eta }{2M1}$, we can return to (A.9) to conclude that if $\Vert U_0\Vert _{\mathbb {H}_0^\tau } < \rho$ we have

$$\begin{aligned} \sup _{t \in [0,T]}\Vert \Upsilon (W)(t)\Vert _{\mathbb {H}_0^\tau } \leqslant M_1\left\{ \rho + \dfrac{2kC_{\omega }}{\tau }\left[ \eta ^2 + \eta ^{3/2}\left( \sup _{t \in [0,T]}\Vert W(t)\Vert _{\mathbb {H}_2^\tau }\right) ^{1/2}\right] \right\} < \eta ,\nonumber \\ \end{aligned}$$

(A.10)

which completes the proof of invariance.

Next, we prove that $\Upsilon$ is a contraction if $\eta$ is sufficiently small. For this, let $W_1 = (w_1,{w_1}_t,{w_1}_{tt})^\top , W_2 = (w_2, {w_2}_{t},{w_2}_{tt})^\top \in X$ and notice that

$$\begin{aligned}&\Vert \Upsilon (W_1) - \Upsilon (W_2)\Vert _{X} =\sup _{t \in [0,T]}\left\| \int _0^t S(t-\sigma ) \left[ F_\tau (W_1)(\sigma )-F_\tau (W_2)(\sigma )\right] d\sigma \right\| _{\mathbb {H}_2^\tau }\nonumber \\&\quad \leqslant \dfrac{C_{\omega }}{\tau } \sup _{t \in [0,T]} \left\| f(w_1)(t)-f(w_2)(t)\right\| _{\mathcal {D}(A^{1/2})}. \end{aligned}$$

(A.11)

Next, observe that for each $t \in [0,T]$—mostly omitted on the computations below—we have

$$\begin{aligned}&(2k)^{-1}\Vert f(w_1)(t) -f(w_2)(t)\Vert _{\mathcal {D}(A^{1/2})}\nonumber \\&\quad =\Vert ({w_1}_t+{w_2}_t)({w_1}_t - {w_2}_{t}) + ({w_1}-{w_2}){w_1}_{tt} + ({w_1}_{tt}-{w_2}_{tt}){w_2}\Vert _{\mathcal {D}(A^{1/2})} \nonumber \\&\quad \leqslant Q_1(t) + Q_2(t) + Q_3(t), \end{aligned}$$

(A.12)

where

$$\begin{aligned} Q_1(t)= & {} \left\| ({w_1}_t+{w_2}_t)({w_1}_t-{w_2}_t)\right\| _{\mathcal {D}(A^{1/2})}, \ \ Q_2(t) = \Vert ({w_1}-{w_2}){w_1}_{tt}\Vert _{\mathcal {D}(A^{1/2})}, \\ Q_3(t)= & {} \Vert ({w_1}_{tt}-{w_2}_{tt}){w_2}\Vert _{\mathcal {D}(A^{1/2})} \end{aligned}$$

and then, we estimate these three quantities (for each $t \in [0,T]$):

$$\begin{aligned}&Q_1(t) \\&\quad = \left\| ({w_1}_t+{w_2}_t)({w_1}_t-{w_2}_t)\right\| _{\mathcal {D}(A^{1/2})} \\&\sim \left\| \nabla \left[ ({w_1}_t+{w_2}_t)({w_1}_t-{w_2}_t)\right] \right\| _{2} \\&\quad = \left\| ({w_1}_t+{w_2}_t)\nabla ({w_1}_t-{w_2}_t)\right\| _{2}+\left\| \nabla \left[ ({w_1}_t+{w_1}_t)\right] ({w_1}_t-{w_2}_t)\right\| _{2} \\&\quad \leqslant \Vert {w_1}_t +{w_2}_t\Vert _{L^\infty }\Vert {w_1}_t - {w_2}_t\Vert _{\mathcal {D}(A^{1/2})} + \Vert \nabla ({w_1}_t + {w_2}_t)\Vert _{L^4}\Vert {w_1}_t - {w_2}_t\Vert _{L^4} \\&\quad \leqslant C\Vert {w_1}_t +{w_2}_t\Vert _{\mathcal {D}(A^{1/2})}^{1/2}\Vert {w_1}_t +{w_2}_t\Vert _{\mathcal {D}(A)}^{1/2}\Vert {w_1}_t - {w_2}_t\Vert _{\mathcal {D}(A^{1/2})} \\&\qquad + C\Vert \nabla ({w_1}_t + {w_2}_t)\Vert _{2}^{1/4}\Vert \nabla ({w_1}_t + {w_2}_t)\Vert _{\mathcal {D}(A^{1/2})}^{3/4}\Vert {w_1}_t - {w_2}_t\Vert _{2}^{1/4}\Vert {w_1}_t - {w_2}_t\Vert _{\mathcal {D}(A^{1/2})}^{3/4} \\&\quad \leqslant C\left[ \Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{1/2}\left[ \Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{1/2}\Vert ({W_1}-{W_2})(t)\Vert _{\mathbb {H}_2^\tau } \\&\qquad +C\left[ \Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{1/4}\left[ \Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/4} \Vert ({W_1}-{W_2})(t)\Vert _{\mathbb {H}_2^\tau } \\&\quad \leqslant C\left[ \sup _{t \in [0,T]}\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{1/2}\left[ \sup _{t \in [0,T]}\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{1/2}\sup _{t \in [0,T]}\Vert ({W_1}-{W_2})(t)\Vert _{\mathbb {H}_2^\tau } \\&\qquad +C\left[ \sup _{t \in [0,T]}\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{1/4}\left[ \sup _{t \in [0,T]}\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/4} \sup _{t \in [0,T]}\Vert ({W_1}-{W_2})(t)\Vert _{\mathbb {H}_2^\tau } \\&\quad \leqslant C\left\{ \eta ^{1/2}\left[ \sup _{t \in [0,T]}\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{1/2} + \eta ^{1/4}\left[ \sup _{t \in [0,T]}\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/4}\right\} \Vert W_1-W_2\Vert _X \\&\quad \leqslant C\eta ^{1/4}\Vert W_1 - W_2\Vert _X, \end{aligned}$$

where C is a constant that does not depend on time.

$$\begin{aligned}&Q_2(t) =\Vert ({w_1}-{w_2}){w_1}_{tt}\Vert _{\mathcal {D}(A^{1/2})} \\&\quad \sim \Vert {w_1}_{tt}\nabla ({w_1}-{w_2}) + ({w_1} - {w_2})\nabla {w_1}_{tt}\Vert _2 \\&\quad \leqslant C\Vert {w_1}_{tt}\Vert _2^{1/4}\Vert {w_1}_{tt}\Vert _{\mathcal {D}(A^{1/2})}^{3/4}\Vert \nabla ({w_1}-{w_2})\Vert _{2}^{1/4}\Vert \nabla ({w_1}-{w_2})\Vert _{\mathcal {D}(A^{1/2})}^{3/4} \\&\qquad + C\Vert {w_1}-{w_2}\Vert _{\mathcal {D}(A^{1/2})}^{1/2}\Vert {w_1}-{w_2}\Vert _{\mathcal {D}(A)}^{1/2}\Vert \nabla {w_1}_{tt}\Vert _2 \\&\quad \leqslant \dfrac{C}{\tau }[\Vert W_1(t)\Vert _{\mathbb {H}_0^\tau }]^{1/4}[\Vert W_1(t)\Vert _{\mathbb {H}_2^\tau }]^{3/4}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau }\\&\qquad +\dfrac{C}{\tau }[\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_0^\tau }]^{1/2}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau } \\&\qquad + \dfrac{C}{\tau } [\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_0^\tau }]^{1/2}[\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau }] \Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau } \\&\quad \leqslant \dfrac{C}{\tau }\left[ \sup _{t \in [0,T]}\Vert W_1(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{1/4}\left[ \sup _{t \in [0,T]}\Vert W_1(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/4}\sup _{t \in [0,T]}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau } \\&\qquad + \dfrac{C}{\tau }\left[ \sup _{t \in [0,T]}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{1/2}\sup _{t \in [0,T]}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau } \\&\qquad + \dfrac{C}{\tau }\left[ \sup _{t \in [0,T]}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_0^\tau }\right] ^{1/2}\\&\quad \left[ \sup _{t \in [0,T]}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] \sup _{t \in [0,T]}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau } \\&\quad \leqslant \dfrac{C}{\tau }\left\{ \eta ^{1/4}\left[ \sup _{t \in [0,T]}\Vert W_1(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/4}+ \eta ^{1/2} + \eta ^{1/2}\left[ \sup _{t \in [0,T]}\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] \right\} \Vert W_1-W_2\Vert _X \\&\quad \leqslant \dfrac{C}{\tau }\eta ^{1/4}\Vert W_1 - W_2\Vert _X, \end{aligned}$$

where C is a constant that does not depend on time.

$$\begin{aligned}&Q_3(t)\\&\quad = \Vert ({w_1}_{tt}-{w_2}_{tt}){w_2}\Vert _{\mathcal {D}(A^{1/2})} \\&\quad \sim \Vert {w_2}\nabla ({w_1}_{tt}-{w_2}_{tt}) + ({w_1}_{tt} - {w_2}_{tt})\nabla {w_2}\Vert _2 \\&\quad \leqslant C\Vert {w_1}_{tt}-{w_2}_{tt}\Vert _2^{1/4}\Vert {w_1}_{tt}-{w_2}_{tt}\Vert _{\mathcal {D}(A^{1/2})}^{3/4}\Vert \nabla {w_2}\Vert _{2}^{1/4}\Vert \nabla {w_2}\Vert _{\mathcal {D}(A^{1/2})}^{3/4} \\&\qquad + C\Vert {w_2}\Vert _{\mathcal {D}(A^{1/2})}^{1/2}\Vert {w_2}\Vert _{\mathcal {D}(A)}^{1/2}\Vert \nabla ({w_1}_{tt}-{w_2}_{tt})\Vert _2 \\&\quad \leqslant \dfrac{C}{\tau }\Big \{[\Vert W_1\Vert _{\mathbb {H}_0^\tau }]^{1/4}[\Vert W_1(t)\Vert _{\mathbb {H}_2^\tau }]^{3/4}\\&\quad + [\Vert (W_1-W_2)(t)\Vert _{\mathbb {H}_0^\tau }]^{1/2}[\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }]^{1/2}\Big \} \Vert ({W_1}-{W_2})(t)\Vert _{\mathbb {H}_2^\tau } \\&\quad \leqslant \dfrac{C}{\tau }\left\{ \eta ^{1/4}\left[ \sup _{t \in [0,T]}\Vert W_1(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{3/4} + \eta ^{1/2}\left[ \sup _{t \in [0,T]}\Vert (W_1+W_2)(t)\Vert _{\mathbb {H}_2^\tau }\right] ^{1/2}\right\} \\&\quad \leqslant \dfrac{C}{\tau }\eta ^{1/4}\Vert W_1 - W_2\Vert _X, \end{aligned}$$

where C is a constant that does not depend on time.

Therefore, back in (A.11), we have

$$\begin{aligned}&\Vert \Upsilon (W_1) - \Upsilon (W_2)\Vert _{X} \nonumber \\&\quad \leqslant \dfrac{C_{\omega }}{\tau } \sup _{t \in [0,T]} \left\| f(w_1)(t)-f(w_2)(t)\right\| _{\mathcal {D}(A^{1/2})} \nonumber \\&\quad \leqslant \dfrac{2kC_{\omega }}{\tau }\sup _{t \in [0,T]}\left[ Q_1(t) + Q_2(t) + Q_3(t)\right] \leqslant \dfrac{2kTC}{\tau ^2}\eta ^{1/4}\Vert W_1-W_2\Vert _X. \end{aligned}$$

(A.13)

which means that $\Upsilon$ is a contraction as long as we take a—possibly smaller—$\eta =\eta (\tau )$ such that $\eta < \left( \dfrac{\tau ^2}{2kC_{\omega }}\right) ^{16}$. This completes the proof of local well-posedness.