On the regularity of solutions to the Moore–Gibson–Thompson equation: a perspective via wave equations with memory

Abstract

We undertake a study of the initial/boundary value problem for the (third order in time) Moore–Gibson–Thompson (MGT) equation. The key to the present investigation is that the MGT equation falls within a large class of systems with memory, with affine term depending on a parameter. For this model equation, a regularity theory is provided, which is also of independent interest; it is shown in particular that the effect of boundary data that are square integrable (in time and space) is the same displayed by the wave equation. Then, a general picture of the (interior) regularity of solutions corresponding to homogeneous boundary conditions is specifically derived for the MGT equation in various functional settings. This confirms the gain of one unity in space regularity for the time derivative of the unknown, a feature that sets the MGT equation apart from other partial differential equation models for wave propagation. The adopted perspective and method of proof enable us to attain as well boundary regularity results for both the integro-differential equation and the MGT equation.

Appendix A: Justification of Definition 3.4

Let us recall that in order to give a Definition of solutions to the MGT Equation (1.2) we proceeded as follows: calculations were used to reduce Eq. (1.2) to the integro-differential Eq. (3.3) and then to the Volterra integral equation (3.13) in the unknown v. By definition, u solves (1.2) when $v(t)=\mathrm{e}^{-(R(0)/2) t} u(t)$ solves the Volterra integral equation (3.13) (with g replaced by $\mathrm{e}^{-(R(0)/2) t}g(t)$). In this Appendix A, we provide a justification of the said Definition.

The argument is similar to the one used in Sect. 2 in the case of wave equations: we prove that the solution u is smooth and can be replaced in both the sides of (1.2) when the initial data and the control are ‘smooth’ and then we use continuous dependence as stated in Table 2 to justify the definition in general. This procedure is a bit more elaborated than the one pertaining to the wave equation, since the third derivative (in time) comes into the picture, which requires more information on the solutions of the wave equation.

In order to distinguish the memoryless wave equation from the equation with memory and the MGT equation, we will denote by $u_3$ the solution to Eq. (1.12). (This is because we use suitable results from [32, § 2.2], where $u_3$ solves the wave equation when the initial data and the affine term are zero.) We assume $u_3(0) \in {{\mathcal {D}}}(\varOmega )$, $\frac{\partial }{\partial t} u_3\big |_{t=0}\in {{\mathcal {D}}}(\varOmega )$, $g\in {{\mathcal {D}}}((0,T) \times \partial \varOmega )$, where ${{\mathcal {D}}}$ denotes the space of $C^\infty$ functions with compact support in the indicated open sets (not be confused with the domain of an operator), while $\partial \varOmega$ is relatively open respect to itself. The assumptions on the affine term F(t) are made explicit below. For the sake of simplicity, in the sequel the time derivative will be denoted by $'$.

It is known that $u_3$ is given by formula (2.2): it is also clear that if $g, f\equiv 0$, then in view of the Sobolev embedding theorems one has $u_3 \in C^\infty ((0,T) \times \varOmega )$, for every $T>0$. Our aim is to show that a similar property holds true when $g\ne 0$, $f\ne 0$.

Let us study separately the effects of g and f: accordingly, we assume first $f=0$, so that

$$\begin{aligned} u_3(t)=-{{\mathcal {A}}}\int \nolimits _0^t R_-(s) Gg(t-s) \,\mathrm{d}s= Gg(t)+ \int \nolimits _0^t R_+(s) Gg'(t-s)\,\mathrm{d}s. \end{aligned}$$

As already noted, we have $u_3(t)-Gg(t)\in \mathcal {D}(A)$ and the boundary condition is satisfied; moreover,

$$\begin{aligned} A \left( u_3(t)-Gg(t)\right) =-Gg''(t)+\int \nolimits _0^t R_+(s) Gg'''(t-s)\, \mathrm {d}s\in C^\infty \big ( [0,T];L^2(\varOmega )\big ) . \end{aligned}$$

Observe that, by definition,

$$\begin{aligned} A \left( u_3(t)-Gg(t)\right) =(\varDelta -I) u_3(t)\in C^\infty \big ( [0,T];L^2(\varOmega )\big ) \end{aligned}$$

that is $u_3(t)\in C^\infty \left( [0,T];H^2(\varOmega )\right)$ with suitable homogeneous boundary condition. Analogously,

$$\begin{aligned} {{\mathcal {A}}}\big [A\big (u_3(t)-Gg(t)\big )+Gg''(t)\big ]=\int \nolimits _0^t R_-(s)Gg^{(4)}(t-s)\, \mathrm {d}s \end{aligned}$$

which again is of class $C^\infty ([0,T];L^2(\varOmega ))$. So we have

$$\begin{aligned} {{\mathcal {A}}}\big [A\big (u_3(t)-Gg(t)\big )+Gg''(t)\big ]\in C^\infty ([0,T];X_1), \end{aligned}$$

that is $u_3\in C^\infty \big ([0,T];H^3(\varOmega )\big )$.

By iteration, we see that in the interior of $(0,T) \times \varOmega$ the solution $u_3$ is of class $C^\infty$, and hence, when computing the derivatives, the order can be interchanged.

Let us consider now the effect of the affine term f(t). We assume $f\in C^\infty \big ([0,T]\times \varOmega \big )$ and that for every fixed $t\ge 0$$f(t,\cdot )\in {{\mathcal {D}}}(\varOmega )$, and yet possibly $f(0,\cdot )\ne 0$.

The contribution of this affine term is

$$\begin{aligned} u_3(t)={{\mathcal {A}}}^{-1}\int \nolimits _0^t R_-(s)f(t-s)\, \mathrm{d}s\in C^\infty \big ([0,T]\times X_1\big ) \end{aligned}$$

since $f^{(n)}(0)\in {{\mathcal {D}}}(A^k)$ for every couple of integers n and k, so that

$$\begin{aligned} u_3(t)\in C^\infty \big ([0,T];X_k\big ) \quad \text {for every }k. \end{aligned}$$

In particular, $u_3 \in C^\infty ([0,T]\times \varOmega )$ as above.

We now extend the obtained properties to the solutions v to the Volterra integral equation (3.13) so that it is possible to track back the computation and to see that equality (1.2) holds pointwise (when the boundary control and the initial conditions have the stated regularity, $u_2\in {{\mathcal {D}}}(\varOmega )$ included).

We confine ourselves to examine the effect of the boundary data g. (The effect of initial data can be examined in a similar way.) Moreover, multiplication by $\mathrm{e}^{-R(0)t/2}$ does not affect the desired results and hence is ignored; i.e. we assume $v(t)\equiv u(t)$.

Because Eq. (3.13) has the form of equation (2.25) in [32, § 2] (the notations are easily adapted, in particular b is substituted by $c^2$ in [32]) following the proof of [32, Theorem 2.4, item 2] we see that $y(t)=v(t)-Gg(t)$ solves

$$\begin{aligned} y(t)=\left( u_3(t)-Gg(t)\right) +\int \nolimits _0^t L(s)Gg(t-s)\,\mathrm{d}s+\int \nolimits _0^t L(s)y(t-s)\,\mathrm{d}s \end{aligned}$$

so that

$$\begin{aligned} {{\mathcal {A}}}y(t)=\int \nolimits _0^t {{\mathcal {A}}}L(s) Gg(t-s)\, \mathrm{d}s+\int \nolimits _0^t L(s){{\mathcal {A}}}y(t-s)\, \mathrm{d}s~ \end{aligned}$$

(note that ${{\mathcal {A}}}L(t)$ is a strongly continuous operator for every t). It then follows that $y(t)\in C^\infty \left( [0,T];X_1\right)$.

Exploiting the definition of L(t) and integrating by parts the integral which contains g(t), we see that $y(t)\in C^\infty \left( [0,T];X_2\right)$. Iterating this procedure, we obtain $u\in C^\infty \left( [0,T]\times \varOmega \right)$. Using this regularity result, we can track back the computation leading to the fact that u(t) solves the MGT equation, including the fact that the Laplacian and the time derivative can be interchanged.