The diffusion phenomenon for dissipative wave equations in metric measure spaces

Abstract

We study the long-time behavior of the solution to a type of dissipative wave equation, where the operator in the equation is time-dependent and the solution is defined on a metric measure space $(X,m)$ satisfying appropriate conditions. The operator is assumed to be self-adjoint and is related to a time-dependent Dirichlet form. We link hyperbolic PDEs with the firmly established theories for parabolic PDEs in metric measure spaces and Dirichlet forms, subsequently deriving the asymptotic behavior of the solution to the dissipative wave equation. We present several nontrivial examples of dissipative wave equations in metric measure spaces where our theory works.

Introduction

Let $(X,m)$ be a separable metric measure space satisfying appropriate conditions, and let $A(t)$ be a time-dependent, densely defined, self-adjoint operator on $L^2(X,m)$. We describe the long-time behavior of the solution to

$$\{\begin{array}{cccc}\hfill & u_{tt}(x,t)+u_t(x,t)+A(t)u(x,t)=0,\hfill & \hfill & x\in X,t>0,\hfill \\ \hfill & (u,u_t)(x,0)=(u_0,u_1)(x),\hfill & \hfill & x\in X,\hfill \end{array}$$

showing that the asymptotic behavior of the solution to (1) aligns with the asymptotic behavior of a prescribed solution of

$$v_t(x,t)+A(t)v(x,t)=0,x\in X,t>0,$$

meaning the solution of (1) exhibits the diffusion phenomenon. We then derive the decay of u, showing that the rate of decay depends on properties of the metric measure space X.

Extensive research has been done to study the behavior of solutions to parabolic PDEs with operators related to general Dirichlet forms; for example, see Sturm [30], Lierl and Saloff-Coste [15] and the references therein. The operator $A(t)$ is related to the time-dependent Dirichlet form ${\mathcal{E}}_t$ via the representation formula ${\mathcal{E}}_t(f,g)={〈A(t)f,g〉}_{L^2(X)}$, which we will discuss later. However, little was known about solutions to hyperbolic PDEs in the context of general Dirichlet forms.

The Dirichlet forms we consider are nonnegative definite, symmetric bilinear forms with domains dense in $L^2(X,m)$. The prototypical Dirichlet form on ${\mathbb{R}}^N$ is $\mathcal{E}(f,g)={\int }_{{\mathbb{R}}^N}a(x)\mathrm{\nabla }f(x)\cdot \mathrm{\nabla }g(x)dx$, where for instance $0\le a(x)\in C^1({\mathbb{R}}^N)$, which corresponds to the self-adjoint operator $\mathbb{A}f=-\mathrm{\nabla }\cdot (a(x)\mathrm{\nabla }f(x))$. In general, Dirichlet forms are definable within the context of many types of metric measure spaces with no relation to partial derivatives. Moreover, in general, $\mathcal{E}(f,g)={\int }_{X}d\mathrm{\Gamma }(f,g)$, where $\mathrm{\Gamma }(f,g)$ is a signed Radon measure on X. Note that the prototypical form satisfies $d\mathrm{\Gamma }(f,g)=a(x)\mathrm{\nabla }f(x)\cdot \mathrm{\nabla }g(x)dx$.

The time-dependent Dirichlet form ${\mathcal{E}}_t(f,g)$ may be interpreted as a collection of Dirichlet forms ${\{{\mathcal{E}}_t\}}_{t\in \mathbb{R}}$ on $L^2(X)$ indexed in t; the formal definition will be given in subsection 1.1. Each ${\mathcal{E}}_t$ corresponds to a nonnegative self-adjoint operator $A(t)$ on $L^2(X)$; see Fukushima, Oshima and Takeda [6, Theorem 1.3.1]. This Theorem gives the presentation ${\mathcal{E}}_t(f,g)={〈A(t)f,g〉}_{L^2(X)}$ for $f\in D(A(t))$ and $g\in D({\mathcal{E}}_t)$, and in many cases, we can rewrite ${〈A(t)f,g〉}_{L^2(X)}$ in a convenient way, allowing us to focus our attention on ${\mathcal{E}}_t$ instead of $A(t)$. This will make our work easier and make our core assumptions less restrictive.

Below, we present several nontrivial examples of dissipative wave equations in metric measure spaces where our theory works. For the precise proofs, see section 7.

• The space X is ${\mathbb{R}}^N$, and we introduce a metric ρ that is intrinsically linked to ${\mathcal{E}}_t$, satisfying $\frac{1}{C}|x-y|\le \rho (x,y)\le C|x-y{|}^{ϵ}$ locally for some $ϵ\in (0,1)$ and $C>0$. This metric ρ is the same as the metric in Fefferman and Phong [3], Fefferman and Sanchez-Calle [4], Jerison and Sanchez-Calle [14], and Nagel, Stein and Wainger [19]. We define the Dirichlet form ${\mathcal{E}}_t(f,g):={\int }_{{\mathbb{R}}^N}{\sum }_{i=1}^N{a}_i(x,t){\partial }_{x_i}f{\partial }_{x_i}gdx$, obtaining the corresponding operator $A(t)$ via the representation $A(t)f(x)=-{\sum }_{i=1}^N{\partial }_{x_i}(a_i(x,t){\partial }_{x_i}f(x))$. The coefficients $a_i(x,t)\ge 0$ are bounded, and the form ${\mathcal{E}}_t$ only needs to satisfy ${\mathcal{E}}_t(f,f)+{\Vert f\Vert }_{L^2({\mathbb{R}}^N)}^2\ge \delta {\Vert f\Vert }_{H^{ϵ}({\mathbb{R}}^N)}^2$ with $\delta >0$ and $ϵ\in (0,1)$ the same as above, independent of t. Hence $A(t)$ is not required to be uniformly elliptic. The coefficient $a_N(x,t)$ in Example E1, given in section 7, is zero for $x=0$.

• The space $(X,m)$ is a (smooth) Riemannian manifold with Riemannian metric d, Riemannian volume measure m and nonnegative Ricci curvature. The self-adjoint operator $A(t)$ satisfies $0\le c_1{\mathrm{\Delta }}_d\le A(t)\le c_2{\mathrm{\Delta }}_d$ for $t\in \mathbb{R}$, where ${\mathrm{\Delta }}_d$ is the Laplace-Beltrami operator defined on X and $c_1,c_2>0$.

• The space X is ${\mathbb{R}}^N$ with a weighted $L^2$ inner product ${〈f,g〉}_{L^2(X)}:={〈f\sqrt{\varphi },g\sqrt{\varphi }〉}_{L^2({\mathbb{R}}^N)}$, where $\varphi (x)\ge 0$. We define the Dirichlet form

  $${\mathcal{E}}_t(f,g):=\underset{{\mathbb{R}}^N}{\int }\sum _{i=1}^N{a}_i(x,t){\partial }_{x_i}f{\partial }_{x_i}g\varphi (x)dx,$$

  obtaining the corresponding operator $A(t)$ via the representation

  $$A(t)f(x)=-\frac{1}{\varphi (x)}\sum _{i=1}^N{\partial }_{x_i}(\varphi (x)a_i(x,t){\partial }_{x_i}f(x)).$$

  For $1\le i\le N$, the coefficients $a_i(x,t)$ are assumed to satisfy $0<a_{i1}\le a_i(x,t)\le a_{i2}$ on ${\mathbb{R}}^N\times \mathbb{R}$, where $a_{i1}$ and $a_{i2}$ are constants.

Particular variations of (1) have been analyzed by many authors, including the initial boundary value problem $u_{tt}+u_t-\mathrm{\Delta }u=0$, where X is a subset of the Euclidean space ${\mathbb{R}}^N$, e.g., see Ikehata [9], [8] and Ono [22] who worked in exterior domains. Also, the preceding IBVP can be considered with a variable damping coefficient, and this problem has been thoroughly studied, e.g., for the space-dependent damping coefficient case see Nakao [20], Ikehata [10], [11], Mochizuki and Nakao [17], Mochizuki and Nakazawa [18], Todorova and Yordanov [34], and Sobajima and Wakasugi [28] who demonstrated the diffusion phenomenon. For the time-dependent damping coefficient case, see Reissig and Wirth [26], and Wirth [35], [36]. Many authors have demonstrated the diffusion phenomenon for the abstract problem $u_{tt}+u_t+Bu=0$ in the Hilbert space $\mathcal{H}$, where B is a time-independent operator. For example, see Ikehata and Nishihara [12], Chill and Haraux [2], Radu, Todorova and Yordanov [24], and Yamazaki [37].

Ikehata, Todorova and Yordanov [13] showed a more complex diffusion phenomenon for abstract wave equations with strong damping. Then Radu, Todorova and Yordanov [25] proved the diffusion phenomenon for the problem $C{u}_{tt}+u_t+Bu=0$ in a Hilbert space $\mathcal{H}$, where B and C are two noncommuting self-adjoint operators on $\mathcal{H}$, which excludes the use of the spectral Theorem. Instead, they used consecutive approximations with conveniently defined diffusion solutions. They also expanded their decay gains that originated in [23], giving the exact gain in the decay rate for $\Vert {\partial }_t^{n}u\Vert$ in terms of $\Vert u\Vert$.

By resolvent arguments, Nishiyama [21] showed the diffusion phenomenon for the problem $u_{tt}+A{u}_t+Bu=0$ in a Hilbert space $\mathcal{H}$. Here A and B are two noncommuting self-adjoint operators on $\mathcal{H}$, satisfying some additional conditions. Yamazaki [37] studied abstract wave equations with time-dependent damping.

Taylor [32] showed the diffusion phenomenon for (1) in the case when X is the Euclidean space ${\mathbb{R}}^N$ and the Dirichlet forms considered are of the type ${\mathcal{E}}_t(f,g)={\int }_{{\mathbb{R}}^N}a(x,t)\mathrm{\nabla }f(x)\cdot \mathrm{\nabla }g(x)dx$, where $a_1\le a(x,t)\le a_2$ for $a_1$ and $a_2>0$. The work in [32] is based on the following three key tools:

The first tool is the improved decay, which specifically refers to the gains in the decay rates for space and time derivatives of u in terms of u. This gain in decay is expressed in a weighted average sense. The improved decay was discussed by Radu, Todorova and Yordanov in [23] and [25].

The weighted energy method developed by Todorova and Yordanov in [33] is the second key tool. A special weight, depending on the particular application being considered, is used. One important consequence of this weighted energy method is that solutions to damped wave equations decay exponentially for x outside of the ball $B_0\left({(t+1)}^{(1+\delta )/2}\right)$, where $\delta >0$ can be arbitrarily small.

The third key tool is the fundamental solution of (2). The fundamental solution of (2) encodes parabolic decay properties; these properties are found in Friedman [5]. We proved that $u-v$, the difference between the solutions of (1) and (2), can be expressed in terms of the fundamental solution of (2) acting on derivatives of u. This presentation of $u-v$ permits the three key tools to work together.

Returning to the problem (1) in the general metric space X, we face several challenges which are exacerbated by the time-dependence in the operator $A(t)$ in (1). In particular, we note that the estimate of Sturm in [30, Corollary 2.5] for the fundamental solution $p(x,t;z,s)$ of the parabolic equation (2) is not sufficient to imply $L^2(X)$ decay for p. Thus, we impose the further condition (S) below to ensure $L^2(X)$ decay for p. Also, a significant amount of effort is required to demonstrate even standard properties such as the finite speed of propagation for the solution to (1), under certain conditions for the operator $A(t)$. Note that there is no finite speed of propagation for the solution to (1) in general metric spaces; see Strichartz [29]. Moreover, we make substantial changes to the proofs of existence and regularity in Lions and Magenes [16, Chapter 3, Section 8] to make them work in our case. We significantly modify the above three key tools of Taylor [32] to make them suitable for our work.

In this subsection, we give assumptions on the time-dependent Dirichlet form that allow us to avoid having to work with the derivative of the operator $A(t)$, which may not be defined. Even if ${\partial }_{t}A(t)$ is defined, it may not be feasible to verify any assumptions that we place on it. Hence, we define derivatives of the time-dependent Dirichlet form ${\mathcal{E}}_t$ and energy measure form ${\mathrm{\Gamma }}_t$ associated with ${\mathcal{E}}_t$.

Let $(X,m)$ be the given separable metric measure space, and let ${\{{\mathcal{E}}_t\}}_{t\in \mathbb{R}}$ be a one-parameter family of (nonnegative) Dirichlet forms, where the domain $D({\mathcal{E}}_t)\subset L^2(X,m)$ is the same for all $t\in \mathbb{R}$. For each fixed $t\in \mathbb{R}$, we assume that ${\mathcal{E}}_t$ is strongly local and regular. By strongly local, we mean ${\mathcal{E}}_t(f,g)=0$ when f is constant on a neighborhood of the support of g. Also, a form is regular exactly when $D({\mathcal{E}}_t)\cap C_c(X)$ is: 1) dense in $D({\mathcal{E}}_t)$ with respect to the norm ${({\mathcal{E}}_t(f,f)+{\Vert f\Vert }_{L^2(X)}^2)}^{1/2}$ and 2) dense in $C_c(X)$ with the uniform norm.

We fix a strongly local, regular Dirichlet form $\mathcal{E}$ on $L^2(X,m)$, and we assume that $D({\mathcal{E}}_t)=D(\mathcal{E})$ for all $t\in \mathbb{R}$. Moreover, we assume that the form $\mathcal{E}$ serves as a reference form, meaning

$$c_1\mathcal{E}(f,f)\le {\mathcal{E}}_t(f,f)\le c_2\mathcal{E}(f,f)$$

for all $f\in D(\mathcal{E})$ and $t\in \mathbb{R}$, where the constants $c_1$ and $c_2>0$. The behavior of this reference form is linked with the behavior of the solution to (2) via the intrinsic metric defined in the next subsection.

Throughout this paper, we will “apply” ${\partial }_t$ to the time-dependent Dirichlet form ${\mathcal{E}}_t$. Hence we define the form derivative, provided it exists,

$$({\partial }_t{\mathcal{E}}_t)(f,g):=\underset{h\to 0}{\lim }({\mathcal{E}}_{t+h}(f,g)-{\mathcal{E}}_t(f,g))/h.$$

We define $({\partial }_t^2{\mathcal{E}}_t)(f,g)$ similarly. We assume that $({\partial }_t^i{\mathcal{E}}_t)\in C(\mathbb{R})$ for $i=1,2$. Now since Dirichlet forms have properties such as a Cauchy-Schwarz type inequality, see Fukushima, Oshima and Takeda [6, Lemma 5.6.1], one might want to make the assumption that $({\partial }_t^i{\mathcal{E}}_t)(f,g)$ is also a Dirichlet form. However, this is restrictive since that would not allow $({\partial }_t^i{\mathcal{E}}_t)(f,f)<0$ for any $f\in D(\mathcal{E})$. To avoid this restriction, we assume that $({\partial }_t^i{\mathcal{E}}_t)$ can be written as a finite linear combination of Dirichlet forms for $i=1,2$, i.e., we assume

$$({\partial }_t^i{\mathcal{E}}_t)(f,g)={\sum }_{j=1}^J{\alpha }_{i,j}{\mathcal{E}}_t^j(f,g)$$

for $i=1,2$ and some $J\in \mathbb{N}$, where ${\alpha }_{i,j}\in \mathbb{R}$ are constants and $|{\alpha }_{1,j}|+|{\alpha }_{2,j}|>0$ for $1\le j\le J$. We assume each form ${\mathcal{E}}_t^j$ is a strongly local, regular Dirichlet form having $D({\mathcal{E}}_t^j)\subset D({\mathcal{E}}_t)$. Note that ${\alpha }_{i,j}$ can be negative.

Remark 1

Many types of Dirichlet forms ${\mathcal{E}}_t$ satisfy assumption (D2). Among the simplest are those where ${\partial }_t^i{\mathcal{E}}_t$ separates into “positive” and “negative” parts, which are again Dirichlet forms. As an example, we consider the Dirichlet forms ${\mathcal{E}}_t(f,g):=K(t)\mathcal{E}(f,g)$ for $K(t)\in C^2(\mathbb{R};\mathbb{R})$, or ${\mathcal{E}}_t(f,g):={\int }_{{\mathbb{R}}^N}a(x,t)\mathrm{\nabla }f(x)\cdot \mathrm{\nabla }g(x)dx$ with $X={\mathbb{R}}^N$ and $a(x,t)\in C^3({\mathbb{R}}^{N+1};\mathbb{R})$ having bounded derivatives. Consequently, to create the “positive” and “negative” parts of ${\partial }_t^i{\mathcal{E}}_t$, we only need to consider the positive and negative parts of ${\partial }_t^{i}K(t)$ or ${\partial }_t^{i}a(x,t)$. Note that the parts of ${\partial }_t^i{\mathcal{E}}_t$ are again Dirichlet forms, in these cases.

Any Dirichlet form $\mathcal{E}$ can be written as $\mathcal{E}(f,g)={\int }_{X}d\mathrm{\Gamma }(f,g)$, where Γ is a nonnegative definite, symmetric bilinear form on $D(\mathcal{E})$ taking values in the signed Radon measures on X. We call Γ the energy measure form associated with $\mathcal{E}$. We will apply ${\partial }_t$ to the energy measure form ${\mathrm{\Gamma }}_t$ associated to the Dirichlet form ${\mathcal{E}}_t$. We will estimate the resulting object ${\partial }_t{\mathrm{\Gamma }}_t$, which we call a form derivation. In particular, we will frequently need local control over the total variation of ${\partial }_t{\mathrm{\Gamma }}_t$. Let ${\mathrm{\Gamma }}_t^j$ by the energy measure forms associated with the Dirichlet forms ${\mathcal{E}}_t^j$ for $1\le j\le J$, where J is the same positive integer as in (D2). For $i=1,2$, define the i-th form derivation of the energy measure form ${\mathrm{\Gamma }}_t$ via

$$({\partial }_t^i{\mathrm{\Gamma }}_t)(f,g):={\sum }_{j=1}^J{\alpha }_{i,j}{\mathrm{\Gamma }}_t^j(f,g),$$

where the ${\alpha }_{i,j}$ are the same as in (D2).

Remark 2

In general, these definitions of form derivations may have nothing to do with derivatives of the energy measure form ${\mathrm{\Gamma }}_t$. However, under convenient assumptions, the form derivations and derivatives of ${\mathrm{\Gamma }}_t$ “coincide”; see Lemma 2.1 below in section 2.

Now for $i=1,2$, we assume

$${\sum }_{j=1}^J|{\alpha }_{i,j}|{\mathrm{\Gamma }}_t^j(f,f)\le \frac{c_{2+i}}{{(|t|+1)}^i}{\mathrm{\Gamma }}_t(f,f),$$

where $c_3,c_4>0$ are constants, $f\in D(\mathcal{E})$ and $t\in \mathbb{R}$. Note that “≤” here means less than or equal to as measures on X.

Remark 3

In sections 3 and 4, we require the full strength of assumption (D3). However, in proving the existence, regularity and finite speed of propagation for the solution u to (1), we can weaken assumption (D3) to ${\sum }_{j=1}^J|{\alpha }_{i,j}|{\mathrm{\Gamma }}_t^j(f,f)\le c_{2+i}{\mathrm{\Gamma }}_t(f,f)$.

Let $\mathcal{E}$ be the reference Dirichlet form coming from the previous subsection, and let Γ be its associated energy measure form. Define $D{(\mathcal{E})}_{loc}:=\{f\in L_{loc}^2(X,m):\mathrm{\Gamma }(f,f)$ is a Radon measure}. The form $\mathcal{E}$ has an associated intrinsic pseudo metric ρ on X defined by

$$\rho (x,y):=sup\{f(x)-f(y):f\in D{(\mathcal{E})}_{loc}\cap C(X),\mathrm{\Gamma }(f,f)\le dm(x)\text{ on }X\}.$$

In general, $\rho (x,y)$ can be infinite or zero for $x\ne y$.

In the case when $X={\mathbb{R}}^N$, m is the Lebesgue measure and $\mathrm{\Gamma }(f,f)=a(x)|\mathrm{\nabla }f(x){|}^2$, we see that ρ is the Euclidean distance d, if $a(x)\equiv 1$. If $a(x)<1$ in some open set U, then ρ may be interpreted as a version of d that has been “stretched” in U, i.e., $d(y,z)\le \rho (y,z)$ for $y,z\in U$. Similarly, if $a(x)>1$ in U, then points in U have been “compressed” together, i.e., $\rho (y,z)\le d(y,z)$.

We assume the topology induced by ρ is equivalent to the original topology on X, guaranteeing that ρ is a metric and $x\mapsto \rho (x,y)$ is continuous on X. We assume that for every $x\in X$ and $R>0$, the ρ-ball $B_R^{\rho }(x):=\{z\in X:\rho (z,x)<R\}$ is relatively compact in X. The relative compactness of the ρ-balls is equivalent to $(X,\rho )$ being complete via Sturm [31, Theorem 2].

Recall that m is the measure given with X. Let $p(x,t;z,s)$ be the fundamental solution of (2), where $x,z\in X$ and $s<t$. We have the estimate from Sturm [30, Corollary 2.5]

$$\begin{array}{cc}\hfill & p{(x,t;z,s)}^{2}m(B_{\sqrt{C_2(t-s)}}^{\rho }(x))m(B_{\sqrt{C_2(t-s)}}^{\rho }(z))\hfill \\ \hfill & \le C_{1}exp(-\frac{\rho {(x,z)}^2}{2C_2(t-s)}){(1+\frac{\rho {(x,z)}^2}{C_2(t-s)})}^J,\hfill \end{array}$$

where constants $C_1,C_2>0$ and $J>2$. We note however that the Gauss-type estimate (3) alone is not sufficient to imply $L^2(X)$ decay for the fundamental solution p of the parabolic equation (2). For R sufficiently large, we make the following assumption on the intrinsic metric ρ:

$${\Vert \underset{X}{\int }m{(B_R^{\rho }(x))}^{-1}m{(B_R^{\rho }(y))}^{-1}exp(-\frac{\rho {(x,y)}^2}{(2+ϵ)R^2})dm(y)\Vert }_{L_x^{\infty }(X)}\le C{R}^{-M},$$

where the constants $ϵ,C$ and $M>0$; we take M to be the largest constant such that (S) holds. As a consequence, we obtain

$${\Vert v(t)\Vert }_{L^2(X)}\le C{(t+1)}^{-\frac{M}{4}}{\Vert v_0\Vert }_{L^1(X)}$$

for $t\ge 1$, where $v(t)$ is a solution to (2) with data $v_0$; see (71) in section 5. In the special case where (2) is $v_t-\mathrm{\Delta }v=0$ with $X={\mathbb{R}}^N$, we see that ρ is the Euclidean metric on ${\mathbb{R}}^N$, meaning (S) is satisfied with $M=N$ if m is the Lebesgue measure. Furthermore, in this case, N is the largest constant satisfying (4) for $t\ge 1$. In section 7, it will be shown that Examples E1 - E3 satisfy (S).

Fix $x_0\in X$ and let $\delta ,\gamma >0$. Define the set $A_{\delta }(t):=\{x\in X:\rho (x,x_0)\ge {(t+1)}^{(1+\delta )/2}\}$ and the function $W_{\gamma }=exp\left(\gamma \frac{\rho {(x,x_0)}^2}{t+1}\right)$. If $(X,\rho ,m)$ were N-dimensional Euclidean space with the Lebesgue measure, then we would have $m(B_R^{\rho }(x_0))\le C{R}^N$ and ${\lim }_{t\to \infty }q(t){\Vert W_{\gamma }^{-1}\Vert }_{L^1(A_{\delta }(t))}=0$ for any polynomial $q(t)$ with real coefficients. Many other triples $(X,\rho ,m)$ also satisfy these two properties. We ask that $(X,\rho ,m)$ is not too “irregular,” which means we assume

$$\underset{t\to \infty }{\lim }q(t){\Vert W_{\gamma }^{-1}\Vert }_{L^1(A_{\delta }(t))}=0\text{and}m(B_R^{\rho }(x_0))\le C{R}^M$$

for some $x_0\in X$, all $\delta ,\gamma >0$, all $R>0$ sufficiently large and all polynomials $q(t)$ with real coefficients. Note that M here is the same constant as in (S). In section 7, it will be shown that Examples E1 - E3 satisfy (NB). We regard $m(B_R^{\rho }(x_0))$ as a loss in the sense that ${\Vert f\Vert }_{L^1(B_R^{\rho }(x_0))}^2\le m(B_R^{\rho }(x_0)){\Vert f\Vert }_{L^2(B_R^{\rho }(x_0))}^2$, but this loss is manageable since we are assuming that M in (NB) is the same as in (S) above.

Let $\mathbb{A}$ be the nonnegative definite, self-adjoint operator associated with the reference Dirichlet form $\mathcal{E}$ from subsection 1.1. We introduce the Hilbert spaces $H:=D(\mathcal{E})\supset V:=D(\mathbb{A})\supset W:=D({\mathbb{A}}^{3/2})$ equipped with their norms respectively

$$\begin{array}{c}\hfill {\Vert f\Vert }_H^2:={\Vert f\Vert }_{L^2(X)}^2+\mathcal{E}(f,f),{\Vert f\Vert }_V^2:={\Vert f\Vert }_H^2+{\Vert \mathbb{A}f\Vert }_{L^2(X)}^2,\hfill \\ \hfill {\Vert f\Vert }_W^2:={\Vert f\Vert }_V^2+{\Vert {\mathbb{A}}^{3/2}f\Vert }_{L^2(X)}^2.\hfill \end{array}$$

Our main results depend on the conclusions of Proposition 1 below, meaning that we are not restricted to the particular hypotheses of Proposition 1; see appendix A for a proof of Proposition 1.

Proposition 1

(Existence and regularity) Let (D1) - (D3) and the assumptions in appendix A.1 be satisfied. Then for any data $u_0\in W$ and $u_1\in V$, there exists a unique solution to (1) such that:

$$0={\Vert A(t)u(\cdot ,t)+u_t(\cdot ,t)+u_{tt}(\cdot ,t)\Vert }_{L^2(X)},$$

$$u_t(t)\in C([0,\infty );H)\cap L^2([0,T];V),$$

$$u_{tt}(t)\in C([0,\infty );L^2(X))\cap L^2([0,T];H),$$

$$u(t)=u_0+\underset{0}{\overset{t}{\int }}{u}_s(s)ds\mathit{\text{and}}{u}_t(t)=u_1+\underset{0}{\overset{t}{\int }}{u}_{ss}(s)ds$$

for all $t\in [0,\infty )$ and all $T>0$.

Since $u_t(t)\in L^2([0,T];V)$ for $T>0$, we have $u(t)\in C([0,\infty );V)$ by the first part of (iv). Also, observe that $u_{tt}$ is an $L^2(X)$ function, not just a distribution. This fact allows us to apply the parabolic theory of Sturm [30].

Remark 4

The standard theory of Lions and Magenes [16, chapter 3, section 8] gives:

$$u(t)\in C([0,\infty );H)\text{ and }{u}_t(t)\in C([0,\infty );L^2(X)).$$

However, we need:

$$u(t)\in C([0,\infty );V)\text{ and }{u}_t(t)\in C([0,\infty );H);$$

obtaining this better regularity solely from [16] requires the distributional identity

$${〈A(t)f,g〉}_H={〈f,A(t)g〉}_H,$$

where $f,g\in V$, which essentially requires the restrictive assumption that $\mathbb{A}$ and $A(t)$ commute for all t. We recover (6) via introducing the time-dependent Hilbert spaces $H(t)$ with common domain $D(\mathcal{E})$ and equivalent norms ${\Vert f\Vert }_{H(t)}^2:={\Vert f\Vert }_{L^2(X)}^2+{\mathcal{E}}_t(f,f)$. Consequently, we have

$${〈A(t)f,g〉}_{H(t)}={〈f,A(t)g〉}_{H(t)}.$$

Improving the regularity given by [16] will most likely require several strong, hard to check assumptions. For instance, we do not expect that the set of eigenfunctions of $A(t)$ remains fixed as t changes.

We assume that (D1) - (D3), the upper Gaussian estimate (3) of Sturm, (S) and (NB) are satisfied. We also assume that the existence and regularity from Proposition 1 hold. Then we obtain the following Theorem and Corollary for data $u_0\in W$ and $u_1\in V$, see (5), with support in $B_{R_0}^{\rho }(w)$ for some $w\in X$ and $R_0>0$:

Theorem 1.1

(Diffusion phenomenon) For $t\ge 0$, the solution $u(x,t)$ to (1) satisfies

$${\Vert u(t)-v(t)\Vert }_{L^2(X)}^2\le C{(t+1)}^{-\frac{M}{2}}{\Vert (u_0,u_1)\Vert }_{V\times H}^2,$$

where $v(t)$ is the solution to (2) prescribed by (71). The constant C depends on $R_0$ and $c_1,c_2,c_3,c_4,M$ coming from (D1) - (D3), (S) and (NB).

Note that M is the same as in conditions (S) and (NB). The prescribed solution $v(t)$ of (2) will be shown to have the decay ${\Vert v(t)\Vert }_{L^2(X)}^2\le C{(t+1)}^{-\frac{M}{2}}{\Vert (u_0,u_1)\Vert }_{V\times H}^2$. Combining this property with Theorem 1.1 gives the following corollary:

Corollary 1

For $t\ge 0$, the solution $u(x,t)$ to (1) satisfies:

$${\Vert u(t)\Vert }_{L^2}^2\le C{(t+1)}^{-\frac{M}{2}}{\Vert (u_0,u_1)\Vert }_{V\times H}^2,$$

$$\mathcal{E}(u(t),u(t))\le C\ln (t+2){(t+1)}^{-\frac{M}{2}-1}{\Vert (u_0,u_1)\Vert }_{V\times H}^2,$$

$${\Vert u_t(t)\Vert }_{L^2}^2\le C\ln (t+2){(t+1)}^{-\frac{M}{2}-2}{\Vert (u_0,u_1)\Vert }_{V\times H}^2,$$

where the constant C depends on $R_0$ and $c_1,c_2,c_3,c_4,M$ coming from (D1) - (D3), (S) and (NB).

This paper is structured as follows. The proof for the finite speed of propagation of the solution u of the wave equation (1) is given in section two. Section three is devoted to the proofs of the improved decay. Estimates coming from the weighted energy method are proved in section four. A representation formula for the difference between solutions of (1) and (2), in terms of the fundamental solution of (2), is derived in section five. In section six, the main result and its corollary are proved via the three key tools. Fully detailed examples are presented in section seven. The proof of existence, uniqueness and regularity for the solution to (1) is given in Appendix A. Appendix B contains proofs for Lemma 2.1, Lemma 2.2 from section 2.