Abstract
We show the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The purpose of the present paper is to formulate and prove results on existence and continuous dependence on parameters of solutions of linear second-order partial differential equations (PDEs) of parabolic type with bounded time delay. To be more specific, consider a rather simplified example, that is, an equation of the form
where \(D \subset \mathbb {R}^N\) is a bounded domain with boundary \(\partial D\), \(\varDelta \) is the Laplace operator in x, \(T > 0\), \(c_1 :(0, T) \times D \rightarrow \mathbb {R}\) belongs to \(L_{\infty }((0, T) \times D)\), and \(R :[0, T] \rightarrow [0, 1]\) is a function in \(L_{\infty }((0, T))\).
The theory of Lyapunov exponents (or rather, more generally, the theory of skew-product dynamical systems) is a powerful tool in the applications of the theory of dynamical systems to the investigation of evolution equations (in a broad sense, containing but not excluded to, ordinary differential equations, parabolic partial differential equations, hyperbolic partial differential equations). That theory requires the (linear) equation to generate a skew-product dynamical system on some bundle whose base is the closure of the set of coefficients of the original equation.
Let us consider two cases. We will remain in the simplified framework of (0.1).
-
The nonautonomous case,
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{\partial u}{\partial t}(t, x) = {\varDelta }u(t, x) + c_1(t, x) u(t - R(t),x), &{} t> 0, \ x \in D \\ u(t, x) = 0 &{} t > 0, \ x \in \partial D, \end{array}\right. } \end{aligned}$$(0.2)where \(c_1\) is defined on \((- \infty , \infty ) \times D\) and R is defined on \((- \infty , \infty )\). We take the closure, in an appropriate topology, of the set of all time-translates of \(c_1\) (the so-called hull). The topology must be, on the one hand, coarse enough for the hull to be a compact (metrizable) space, and, on the other hand, fine enough for, first, the time translation operator on the hull to be continuous, and, second, the solution operator to depend continuously on parameters, that is, members of the hull. The paper [39] gives a survey of subsets of function spaces that can serve as hulls. For the theory of linear skew-product (semi)flows on bundles whose fibers are Banach spaces and some of its applications, see, e.g., [4, 10,11,12, 35, 36, 40, 41] for a very incomplete list arranged in chronological order.
-
The random case,
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{\partial u}{\partial t}(t, x) = {\varDelta }u(t, x) + c_1(\theta _{t}\omega , x) u(t - R(\theta _{t}\omega ),x), &{} t> 0, \ x \in D \\ u(t, x) = 0 &{} t > 0, \ x \in \partial D, \end{array}\right. } \end{aligned}$$(0.3)where \(c_1\) is now defined on \(\varOmega \times D\) and R is defined on \(\varOmega \), with \((\varOmega , {\mathfrak {F}}, \mathbb {P})\) a probability space on which an ergodic measurable flow \(\theta = (\theta _t)_{t \in \mathbb {R}}\) acts. Here the role of hull is played by \(\varOmega \), and the measurability of the flow \(\theta \) is one of the assumptions. In order to apply the theory of Lyapunov exponents in the measurable setting, as presented in, e.g., [9, 25,26,27, 30], one needs to show the measurable dependence of the solution operators on \(\omega \in \varOmega \)
In the present paper, we address the problem of continuous dependence on members of the hull. As the space of coefficients, we take a closed and bounded subset of the Banach space of essentially bounded (Lebesgue-)measurable functions on \((0, T) \times D\), where \(T > 0\), with the weak-* topology induced by the duality pairing between \(L_1\) and \(L_{\infty }\). Regarding the zero-order coefficients and delay terms, no additional assumption is made. In particular, the dependence on t can be quite weak.
Although there have been a lot of papers dealing with the issues of the existence of solutions of delay PDEs (many of them nonlinear, and admitting more general delay terms, employing various definitions of solutions, see, e.g., [6,7,8, 20, 21, 23, 24, 31, 32, 42,43,44, 46]), the only papers we are aware of dealing explicitly with continuous dependence of solutions of delay PDEs on parameters are [37] and [38].
To give a flavor of our results, we formulate now some specializations of our main results to the case of (0.1). We assume \(1< p < \infty \).
The first, a specialization of Theorem 3.1, establishes the existence and uniqueness of mild solutions.
Theorem
Let \(c_1 \in L_{\infty }((0, T) \times D)\), \(u_0 \in C([-1, 0], L_p(D))\) and \(R \in L_{\infty }((0, T))\) be such that \(R(t) \in [0, 1]\) for Lebesgue-a.e. \(t \in (0, T)\). Then there exists a unique solution \(u(\cdot ; c_1, u_0, R) \in C([-1, T])\) of Eq. (0.1) with initial condition \(u(t; c_1, u_0, R) = u_0\), \(t \in [-1, 0]\). The solution is understood in a suitable integral sense (a mild solution).
The second, a specialization of Theorem 5.1(ii), establishes the continuity, in a suitable sense, of a solution with respect to initial conditions and parameters.
Theorem
Assume that \((c_{1,m})_{m = 1}^{\infty }\), \((u_{0,m})_{m = 1}^{\infty }\) and \((R_m)_{m = 1}^{\infty }\) are sequences satisfying the following:
-
\(c_{1,m}\) have their \(L_{\infty }((0, T) \times D)\)-norms uniformly bounded, and converge in the weak-* topology to \(c_1 \in L_{\infty }((0, T) \times D)\);
-
\(u_{0,m}\) converge in the norm topology of \(C([-1, 0], L_p(D)\) to \(u_0\);
-
\(R_m\) converge for Lebesgue-a.e. \(t \in (0, T)\) to R.
Then,
in the \( C([-1, T], L_p(D))\)-norm.
The paper is organized as follows.
Section 1 presents the assumptions used throughout.
In Sect. 2, results concerning the existence and basic properties of (weak) solutions to linear parabolic PDEs without delay terms are gathered. They are for the most part taken from [35] and based on [13], though some of them (Proposition 2.18, for example), perhaps belonging to the folk lore, appear in print for the first time.
Section 3 is devoted to defining and proving the existence and uniqueness of (mild) solutions of PDEs with delay terms. Section 4 provides estimates of the solutions which are then used to prove the continuous dependence on initial conditions.
Section 5 can be considered the main part of the paper. Here, the continuous dependence of solutions on coefficients and delay terms is proved under very weak assumptions: coefficients are required to converge in the weak-* topology only.
It should be mentioned that a similar approach has been successfully applied in the case of ordinary differential equations with delay in [33, 34, 36], see also [5, 17, 18, Chpt. 5].
1.1 General notations
We write \(\mathbb {R}^{+}\) for \([0, \infty )\), and \(\mathbb {Q}\) for the set of all rationals.
If \(B \subset A\), we write \(\mathbb {1}_{\! B}\) for the indicator of B: \(\mathbb {1}_{\! B}(a) = 1\) if \(a \in B\) and \(\mathbb {1}_{\! B}(a) = 0\) if \(a \in A \setminus B\).
For a metric space (Y, d), \({\mathfrak {B}}(Y)\) denotes the \(\sigma \)-algebra of all Borel subsets of Y.
For Banach spaces \(X_1\), \(X_2\) with norms \(\Vert \cdot \Vert _{X_1}\), \(\Vert \cdot \Vert _{X_2}\), we let \({\mathcal {L}}(X_1, X_2)\) stand for the Banach space of bounded linear mappings from \(X_1\) into \(X_2\), endowed with the standard norm \(\Vert \cdot \Vert _{X_1, X_2}\). Instead of \({\mathcal {L}}(X, X)\) we write \({\mathcal {L}}(X)\), and instead of \(\Vert \cdot \Vert _{X, X}\) we write \(\Vert \cdot \Vert _{X}\). \({\mathcal {L}}_{\textrm{s}}(X_1, X_2)\) denotes the space of bounded linear mappings from \(X_1\) into \(X_2\) equipped with the strong operator topology. Instead of \({\mathcal {L}}_{\textrm{s}}(X, X)\), we write \({\mathcal {L}}_{\textrm{s}}(X)\).
Throughout the paper, \(T > 0\) will be fixed.
We set
Throughout the paper, \(D \subset \mathbb {R}^N\) stands for a bounded domain, with boundary \(\partial D\).
By \({\mathfrak {L}}((0, T))\), we understand the \(\sigma \)-algebra of all Lebesgue-measurable subsets of (0, T). The notations \({\mathfrak {L}}(D)\) and \({\mathfrak {L}}((0, T) \times D)\) are defined in a similar way.
For u belonging to a Banach space of (equivalence classes of) functions defined on D, we will denote by u[x] the value of u at \(x \in D\).
\(L_p(D)=L_p(D,\mathbb {R})\) has the standard meaning, with the norm, for \(1 \le p < \infty \), given by
and for \(p = \infty \) given by
For \(1 \le p \le \infty \) let \(p'\) stand for the Hölder conjugate of p. The duality pairing between \(L_p(D)\) and \(L_{p'}(D)\) is given, for \(1< p < \infty \), or for \(p =1\) and \(p' = \infty \), by
Let u be an equivalence class of functions defined for Lebesgue-a.e. \(t \in (0, T)\) and taking values in \(L_p(D)\), \(1 \le p < \infty \) (in the sequel we will refer to such u simply as a function).
-
u is said to be measurable if it is \(({\mathfrak {L}}((0, T)), {\mathfrak {B}}(L_p(D)))\)-measurable, meaning that the preimage under u of any open subset of \(L_p(D)\) belongs to \({\mathfrak {L}}((0, T))\).
-
u is strongly measurable (sometimes called Bochner measurable) if there exists a sequence \((u_m)_{m = 1}^{\infty }\) of simple functions such that \(\lim \nolimits _{m \rightarrow \infty } \Vert u_m(t) - u(t) \Vert _{L_p(D)} = 0\) for Lebesgue-a.e. \(t \in (0, T)\).
-
u is weakly measurable if for any \(v \in L_{p'}(D)\) the function
$$\begin{aligned}{}[\, t \mapsto \langle u(t), v \rangle _{L_p(D), L_{p'}(D)} \,] \end{aligned}$$is \(({\mathfrak {L}}((0, T)), {\mathfrak {B}}(\mathbb {R}))\)-measurable.
Theorem 0.1
For \(u :(0, T) \rightarrow L_p(D)\) measurability, strong measurability and weak measurability are equivalent.
The equivalence of strong and weak measurability is a consequence of Pettis’s measurability theorem (see, e.g., [16, Thm. 2.1.2]). For the fact that measurability implies strong measurability see, e.g., [45, Thm. 1], whereas the proof of the reverse implication is a simple exercise.
For our purposes, we will use the following definitions (see, e.g., [2, Sect. X.4]). A measurable \(u :(0, T) \rightarrow L_p(D)\) belongs to \(L_r((0, T), L_p(D))\), \(1 \le r < \infty \), if \(\Vert u(\cdot )\Vert _{L_p(D)}\) belongs to \(L_r((0, T))\), with
Similarly, a measurable \(u :(0, T) \rightarrow L_p(D)\) belongs to \(L_{\infty }((0, T), L_p(D))\), if \(\Vert u(\cdot )\Vert _{L_p(D)}\) belongs to \(L_{\infty }((0, T))\), with
The following result, a part of [19, Lemma III.11.16], will be used several times.
Lemma 0.1
-
(a)
If \(u \in L_1((0, T), L_1(D))\) then the function
$$\begin{aligned} \bigl [\, (0, T) \times D \ni (t, x) \mapsto u(t)[x] \in \mathbb {R}\,\bigr ] \end{aligned}$$belongs to \(L_1((0, T) \times D, \mathbb {R})\).
-
(b)
If w is \(({\mathfrak {L}}((0, T) \times D), {\mathfrak {B}}(\mathbb {R}))\)-measurable, and for Lebesgue-a.e. \(t \in (0, T)\) the t-section \(w(t, \cdot )\) belongs to \(L_p(D)\), where \(1 \le p < \infty \), then the function
$$\begin{aligned} \Bigl [\, (0, T) \ni t \mapsto \bigl [\, D \ni x \mapsto w(t, x) \in \mathbb {R}\,\bigr ] \, \Bigr ] \end{aligned}$$is \(({\mathfrak {L}}(0, T), {\mathfrak {B}}(L_p(D)))\)-measurable.
Remark 0.1
Regarding Lemma 0.1(b), we remark that in [19] the analog of w is assumed to be \(({\mathfrak {L}}((0, T)) \otimes {\mathfrak {L}}(D),{\mathfrak {B}}(\mathbb {R}))\)-measurable (no completion) rather than \(({\mathfrak {L}}((0, T) \times D),{\mathfrak {B}}(\mathbb {R}))\)-measurable. As an \(({\mathfrak {L}}((0, T) \times D), {\mathfrak {B}}(\mathbb {R}))\)-measurable function can be made into an \(({\mathfrak {L}}((0, T)) \otimes {\mathfrak {L}}(D), {\mathfrak {B}}(\mathbb {R}))\)-measurable function by changing its values on a set of \((N + 1)\)-dimensional Lebesgue measure zero (see, e.g., [22, Prop. 2.12]), our formulation follows.
2 Assumptions and definitions
2.1 Main equation
Consider a linear second-order partial differential equation with bounded delay
The delay map \(R :[0,T]\rightarrow \mathbb {R}\) is bounded from below by 0 and from above by 1, i.e.,
Sometimes the function \(\xi \mapsto \xi -R(\xi )\) will be denoted by \(\Phi \). The function \(\Phi \) will be called relative time delay. Further, \(D \subset \mathbb {R}^N\) is a bounded domain with boundary \(\partial D\). The equation (ME) will be complemented with boundary conditions
Later on, we will use the notation \({\mathcal {B}}_a\) in other to exhibit dependence of the operator \({\mathcal {B}}\) on a. The boundary conditions operator (BC) will be one of this form
The vector \(\varvec{\nu } = (\nu _{1},\dots ,\nu _{N})\) denotes the unit normal on the boundary \(\partial D\) pointing out of D, interpreted in a certain weak sense (in the regular sense if \(\partial D\) is sufficiently smooth [35]).
The initial condition is considered in the following way: for \(u_0 \in C([- 1, 0], L_p(D))\), where \(1 \le p \le \infty \), find a solution of (ME)\(\,+\,\)(BC) satisfying
By (ME)\(\,+\,\)(BC) we understand equation (ME) equipped with boundary condition (BC). Later on, we will also use \((ME)_a+(BC)_a\) notation to indicate that parameters of ((ME) + (BC)) are fixed to be a.
Note that, without any additional assumptions on the delay map R, the initial data cannot be taken from \(L_p(D) \oplus L_r((- 1, 0), L_p(D))\), as in [33] or [34]. The reason for this is that the delay map R can be constructed in such way that \(t\mapsto t-R(t)\) would be a constant function. In such a situation, the initial value problem ((ME) + (BC)) would be not meaningful. Under some additional assumptions, the situation can change; for example, a constant delay map allows us to introduce generalized initial data in \(L_p(D) \oplus L_r((- 1, 0), L_p(D))\). However, in this paper we will not focus on that.
In order to clearly define the problem \((ME)_{a}+(BC)_a\), it is also necessary to set the delay map R. However, the assumptions on R will be given later. Moreover, we suppress the notation of R from \((ME)_{a}+(BC)_a\). We will present the solutions of \((ME)_{a}+(BC)_a\) in the form of \(u(\cdot ;a,u_0,R)\) and often suppress the notation of \(a,u_0\) or R if it does not lead to confusion.
2.2 Main assumptions
We introduce some assumptions on the domain \(D\subset \mathbb {R}^N\) and the coefficients of the problem (ME) + (BC).
- (DA1):
-
(Boundary regularity) For Dirichlet boundary conditions, D is a bounded domain. For Neumann or Robin boundary conditions, D is a bounded domain with Lipschitz boundary.
In all expressions of the type “a.e.” we consider 1-dimensional Lebesgue measure on (0, T), N-dimensional Lebesgue measure on D and \((N - 1)\)-dimensional Hausdorff measure on \(\partial D\). The latter is, by (DA1), equal to surface measure on \(\partial D\).
The notation \(L_{\infty }(\partial D)\) [resp. \(L_{\infty }((0, T) \times \partial D)\)] corresponds to surface measure on \(\partial D\) [resp. to the product of 1-dimensional Lebesgue measure on (0, T) and surface measure on \(\partial D\)].
- (DA2):
-
(Boundedness) The functions
- \(\diamond \):
-
\(a_{ij} :(0,T) \times D \rightarrow \mathbb {R}\) (\(i, j = 1, \dots , N\)),
- \(\diamond \):
-
\(a_{i} :(0,T) \times D \rightarrow \mathbb {R}\) (\(i = 1, \dots , N\)),
- \(\diamond \):
-
\(b_{i} :(0,T) \times D \rightarrow \mathbb {R}\) (\(i = 1, \dots , N\)),
- \(\diamond \):
-
\(c_0 :(0,T) \times D \rightarrow \mathbb {R}\),
- \(\diamond \):
-
\(c_1 :(0,T) \times D \rightarrow \mathbb {R}\)
belong to \(L_{\infty }((0,T)\times D)\). When the Robin boundary condition holds the function \(d_0 :(0,T) \times \partial D \rightarrow \mathbb {R}\) belongs to \(L_{\infty }((0,T)\times \partial D)\).
It is worth noticing at this point that uniform \(L_{\infty }(D)\)-boundedness of \(a_{ij}(t,\cdot )\), \(a_{i}(t,\cdot )\), \(b_{i}(t,\cdot )\), \(c_{0}(t,\cdot )\), \(c_1(t,\cdot )\) and uniform \(L_{\infty }(\partial D)\)-boundedness of \(d_0(t,\cdot )\) for a.e. \(t\in (0,T)\) follow from the assumption (DA2) and Fubini’s theorem.
Definition 1.1
(Y coefficients space). Let Y be a subset of the Banach space \(L_{\infty }((0,T) \times D, \mathbb {R}^{N^2 + 2N +2}) \oplus L_{\infty }((0,T) \times \partial D, \mathbb {R})\) satisfying the following assumptions
-
(Y1)
Y is norm-bounded and, moreover, it is closed (hence compact, via the Banach–Alaoglu theorem) in the weak-* topology,
-
(Y2)
the function \(d_0 \ge 0\) if the Robin boundary condition holds. The function \(d_0\) is interpreted as the zero function in the Dirichlet or Neumann cases.
Elements of Y will be denoted by
The weak-* topology of the space Y is understood in the standard sense, namely, as the weak-* topology induced via the isomorphism
Definition 1.2
(Flattening Y to \(Y_{0}\)). The mapping defined on Y by
will be called the flattening of \(a\in Y\).
The above mapping is obviously continuous. As a consequence, the image \(Y_0\) of Y under that mapping shares properties analogous to (Y1) and (Y2).
-
(DA3)
(Ellipticity) There exists a constant \(\alpha _0 > 0\) such that for any \(a_0 \in Y_0\) the inequality
$$\begin{aligned} \displaystyle \sum \limits _{i,j=1}^{N} a_{ij}(t, x) \xi _{i} \xi _{j} \ge \alpha _0 \sum \limits _{i=1}^{N} \xi _{i}^2, \end{aligned}$$holds for a.e. \((t,x) \in (0,T) \times D\) and all \(\xi \in \mathbb {R}^{N}\), and the functions \(a_{ij}(\cdot ,\cdot )\) are symmetric in the indices, i.e., \(a_{ij}(\cdot ,\cdot ) \equiv a_{ji}(\cdot ,\cdot )\) for all \(i, j = 1, \dots , N\).
-
(DA4)
(Sequential compactness of \(Y_0\) with respect to convergence a.e.) Any sequence \((a_{0,m})_{m = 1}^{\infty }\) of elements of \(Y_0\), where
$$\begin{aligned} a_{0, m} := \bigl ( (a_{ij,m})_{i,j=1}^{N}, (a_{i,m})_{i=1}^{N}, (b_{i,m})_{i=1}^{N}, c_{0,m}, 0, d_{0,m} \bigr ), \end{aligned}$$convergent as \(m \rightarrow \infty \) in the weak-* topology to \(a_0 \in Y_0\) has the property that
-
the sequence \(\bigl ( (a_{ij,m})_{i,j=1}^{N}, (a_{i,m})_{i=1}^{N}, (b_{i,m})_{i=1}^{N} \bigr )\) converges to \(\bigl ( (a_{ij})_{i,j=1}^{N}, (a_{i})_{i=1}^{N}, (b_{i})_{i=1}^{N} \bigr )\) pointwise a.e. on \((0,T) \times D\),
-
the sequence \(d_{0,m}\) converges to \(d_{0}\) pointwise a.e. on \((0,T) \times \partial D\).
Occasionally, we will use the following.
-
-
(DA5)
\(Y_0\) is a singleton.
For the purposes of studying continuous dependence on parameters and delay, we introduce now the delay class and the relative delay class equipped with suitable topologies.
Definition 1.3
The delay class is defined as follows
The delay class is equipped with the weak-* topology.
At some moments we also use the following notation \(\widetilde{{\mathcal {R}}} := \{[\,t\mapsto t-R(t)\,]: \, R\in {\mathcal {R}} \}\) especially in more abstract lemmas when general properties of the mapping \(t\mapsto t-R(t)\) are important.
Remark 1.1
The delay class \({\mathcal {R}}\) is a norm-bounded, convex and weak-* closed subset of \( L_{\infty }((0,T))\), hence, by the Banach–Alaoglu theorem, it is compact in the weak-* topology.
The following assumption is a property of a subset \({\mathcal {R}}_0\subset {\mathcal {R}}\).
-
(DA6)
If \(R\in {\mathcal {R}}_0\) is a weak-* limit of \((R_m)_{m=1}^{\infty }\subset {\mathcal {R}}_0\) then \((R_m)_{m=1}^{\infty }\) converge pointwise a.e. on (0, T) to R.
Remark 1.2
Note that the assumption (DA6) is naturally satisfied when \({\mathcal {R}}_0\) is compact in \({\mathcal {R}}\) with respect to the norm topology. This fact follows from an observation that any Hausdorff topology weaker than the norm topology (such as the weak-* topology) is equal to the norm topology on a compact subset.
Remark 1.3
Note that the weak-* topology on Y is metrizable, see [35, (1.3.1)]. Similarly, the weak-* topology on \({\mathcal {R}}\) is metrizable, see [14, Thm. 3.6.17 and Cor. 3.6.18].
3 Weak solutions
In the present section, we assume (DA1), (DA2) and that the flattening \(Y_0\) of Y as in Definition 1.1 satisfies (DA3). Occasionally we will assume (DA4).
We start with a PDE parameterized by \(a_0 \in Y_{0}\)
Equation (\(\widehat{\textrm{ME}}\)) is complemented with boundary conditions
where \({\mathcal {B}}\) is either the Dirichlet or the Neumann or else the Robin boundary operator.
We are looking for solutions of the problem ((\(\widehat{\textrm{ME}}\)))\(_{a_0}\)+ ((\(\widehat{\textrm{BC}}\)))\(_{a_0}\) for initial condition \(u_0\in L_2(D)\). To define a solution we introduce the space H as follows. Let
and
equipped with the norm
where \({\dot{v}} := \textrm{d}v/\textrm{d}t\) is the time derivative in the sense of distributions taking values in \(V^{*}\) (see [15, Chpt. XVIII] for definitions).
For \(a_0 \in Y_{0}\) define a bilinear form
in the Dirichlet or Neumann case, and
in the Robin case, where \(H_{N - 1}\) stands for the \((N-1)\)-dimensional Hausdorff measure.
Definition 2.1
(Local Weak Solution). For \(a_0 \in Y_{0}\), \(0 \le s\le t\le T\) and \(u_0 \in L_2(D)\) a function \(u \in L_2([ s, t ],V)\) such that \({\dot{u}} \in L_2( [ s, t ], V^{*} )\) is a weak solution of ((\(\widehat{\textrm{ME}}\)))\(_{a_0}\)+ ((\(\widehat{\textrm{BC}}\)))\(_{a_0}\) on [s, t] with initial condition \(u(s) = u_0\) if
for any \(v \in V\) and any \(\psi \in {\mathcal {D}}([s, t),\mathbb {R})\) where set \({\mathcal {D}}([s, t), \mathbb {R})\) is the space of all smooth real functions having compact support in [s, t) and \((\cdot , \cdot )_{L_2(D)}\) denotes the standard inner product in \(L_2(D)\).
Definition 2.2
(Global Weak Solution). When \(t=T\) in definition 2.1, a weak solution will be called a global weak solution.
Proposition 2.1
(Existence of global weak solution) For any initial condition \(u_0\in L_2(D)\), there exists a unique global weak solution of (\(\widehat{\textrm{ME}}\))\(\,+\,\)(\(\widehat{\textrm{BC}}\)).
Proof
See [13, Thm. 2.4] for a proof and [35, Prop. 2.1.5] for a unified theory of weak solutions. \(\square \)
For \(a_0\in Y_0\) and \(0\le s<T\), we write the unique global weak solution of ((\(\widehat{\textrm{ME}}\)))\(_{a_0}\)+ ((\(\widehat{\textrm{BC}}\)))\(_{a_0}\) with initial condition \(u(s)=u_0\) as \(U_{a_0} (t,s)u_0:=u(t)\).
Below we present a couple of results from [35, Ch. 2].
Proposition 2.2
The mappings
have the following properties.
Proof
See [35, Props. 2.1.5 through 2.1.8]. \(\square \)
Proposition 2.3
-
(i)
Let \(1 \le p < \infty \) and \(0 \le s < T\). For any \(a_0 \in Y_0\) there exists \(U_{a_{0},p}(t) \in {\mathcal {L}}(L_p(D))\) such that
$$\begin{aligned} U_{a_{0},p}(t,s) u_0 = U_{a_{0}}(t,s) u_0, \quad u_0 \in L_2(D) \cap L_p(D). \end{aligned}$$ -
(ii)
Let \(1< p < \infty \) and \(a_{0} \in Y_0\). Then, the mapping
$$\begin{aligned} \bigl [\, [s, T] \ni t \mapsto U_{a_{0},p}(t,s) \in {\mathcal {L}}_{\textrm{s}}(L_p(D)) \,\bigr ] \end{aligned}$$is continuous.
Proof
See [13, Cor. 7.2] for part (i) and [13, Thm. 5.1] for part (ii). \(\square \)
For \(p=1\), we have an analog of Proposition 2.3(ii).
Proposition 2.4
Let \(1 \le p < \infty \), \(0 \le s < T\) and \(a_0 \in Y_0\). Then, the mapping
is continuous.
Proof
See [35, Prop. 2.2.6]. \(\square \)
For \(0 \le s < T\), we write \(U_{a_0,p}(s,s) = \textrm{Id}_{L_p(D)}\) even if \(p = 1, \infty \).
Proposition 2.5
For any \(a_0\in Y_{0}\), \(0 \le s\le t_1\le t_2\le T\) and any \(1\le p\le \infty \)
Proof
See [35, Prop. 2.1.7] for the proof of \(p=2\) case. For \(p\not =2\) it suffices to use the fact that \(U_{a_0}(t,s)\in {\mathcal {L}}(L_2(D))\) and the continuity of the mappings \([u\mapsto U_{a_0,p}(t_2,t_1)\circ U_{a_0,p} (t_1,s)u]\) and \([u\mapsto U_{a_0,p}(t_2,s)u]\), which is guaranteed by Proposition 2.3. \(\square \)
Proposition 2.6
For any \(a_0\in Y_{0}\) and any \(0 \le s\le t_1\le t_2\le T\) the operator \( U_{a_0}(t_2,t_1)\) has an a.e. nonnegative kernel.
Proof
See [3, Thm. 1.3] for the existence of a kernel, for nonnegativity see [13, Cor. 8.2]. \(\square \)
Proposition 2.7
-
(i)
For any \(a_{0} \in Y_0\), any \((s,t) \in \dot{\varDelta }\) and any \(1 \le p \le q \le \infty \) there holds \(U_{a_{0}}(t,0) \in {\mathcal {L}}(L_p(D), L_q(D))\).
-
(ii)
There are constants \(M \ge 1\) and \(\gamma \in \mathbb {R}\) such that
$$\begin{aligned} \left\| U_{a_0}(t, s)\right\| _{{\mathcal {L}}(L_{p}(D), L_{q}(D))} \le M (t-s)^{-\frac{N}{2}\left( \frac{1}{p}-\frac{1}{q}\right) } e^{\gamma (t-s)} \end{aligned}$$(2.4)for \(1 \le p \le q \le \infty \), \(a_{0} \in Y_0\) and \((s,t)\in \dot{\varDelta }\).
Proof
See [13, Sect. 5 and Cor. 7.2]. \(\square \)
In particular, setting \(p = q\) we have
In the sequel we will frequently assume that \(\gamma \ge 0\) in Proposition 2.7 and its derivatives.
Proposition 2.8
Let \(1 \le p \le \infty \) and \(0 \le s < T\). Then, for any \(T_1 \in (s, T]\) there exists \(\alpha \in (0,1)\) such that for any \(a_0 \in Y_0\), any \(u_0 \in L_p(D)\), and any compact subset \(D_0 \subset D\) the function \(\bigl [\,[T_1,T] \times D_0 \ni (t,x) \mapsto (U_{a_0}(t) u_0)[x]\,\bigr ]\) belongs to \(C^{\alpha /2,\alpha }([T_1,T] \times D_0)\). Moreover, for fixed \(T_1\), and \(D_0\), the \(C^{\alpha /2,\alpha }([T_1,T] \times D_0)\)-norm of the above restriction is bounded above by a constant depending on \(\Vert u_0\Vert _{L_p(D)}\) only.
Proof
It follows from Proposition 2.7 and from [29, Chpt. III, Thm. 10.1]. \(\square \)
Proposition 2.9
For any \((s,T_1)\in \dot{\varDelta }\), \(1 \le p < \infty \) and a bounded \(E \subset L_p(D)\) the set
is precompact in \(C([T_1, T], L_p(D))\).
Proof
Fix \((s,T_1)\in \dot{\varDelta }\), \(1 \le p < \infty \) and a bounded \(E \subset L_p(D)\). Let \((a_{0,m})_{m=1}^{\infty } \subset Y_0\) and \((u_{0,m})_{m=1}^{\infty } \subset E\). Put, for \(m = 1, 2, \dots \),
It follows from Proposition 2.8 via the Ascoli–Arzelà theorem by diagonal process that, after possibly taking a subsequence, \((u_m)_{m = 1}^{\infty }\) converges as \(m \rightarrow \infty \) to some function \({\tilde{u}}\) defined on \([T_1, T]\) and taking values in the set of continuous real functions on D in such a way that for any compact \(D_0 \subset D\) the functions \([\, t \mapsto u_m(t)\!\!\restriction _{D_0} \,]\) converge to \([\, t \mapsto {\tilde{u}} (t)\!\!\restriction _{D_0} \,]\) in \(C([T_1, T], C(D_0))\).
We claim that \(u_m\) converge to \({\tilde{u}}\) in the \(C([T_1, T], L_p(D))\)-norm. By Proposition 2.7, there is \(M > 0\) such that \(\Vert u_m(t)\Vert _{L_{\infty }(D)} \le M\) and \(\Vert {\tilde{u}}(t)\Vert _{L_{\infty }(D)} \le M\) for all \(m = 1, 2, \ldots \) and all \(t \in [T_1, T]\). For \(\epsilon > 0\) take a compact \(D_0 \subset D\) such that \(\lambda (D \setminus D_0) < (\epsilon /(4 M))^p\), where \(\lambda \) denotes the N-dimensional Lebesgue measure. We have
for all \(m = 1, 2, \ldots \) and all \(t \in [T_1, T]\). Further, since \([\, t \mapsto u_m(t)\!\!\restriction _{D_0} \,]\) converge to \( [\, t \mapsto {\tilde{u}}(t)\!\!\restriction _{D_0} \,]\) in the \(C([T_1, T], C(D_0))\)-norm, there is \(m_0\) such that
for all \(m \ge m_0\) (here \(\mathbb {1}_{\! D_0}\) stands for the function constantly equal to \(\mathbb {1}_{\! D_0}\)). Consequently,
for all \(m \ge m_0\). \(\square \)
Corollary 2.1
Let \(1 \le p < \infty \), \(0 \le s < T\), \(a_0 \in Y_0\). Then the mapping
is continuous.
Proof
Let \((u_m)_{m = 1}^{\infty }\) converge in \(L_p(D)\) to \(u_0\) and let \((t_m)_{m = 1}^{\infty }\) converge to \(t > s\). Take \(\epsilon > 0\). It follows from Proposition 2.4 that there is \(m_1\) such that \(\Vert U_{a_0}(t_m, s) u_0 - U_{a_0}(t, s) u_0\Vert _{L_p(D)} < \epsilon /2\) for \(m \ge m_1\), and it follows from Proposition 2.7(ii) that there is \(m_2\) such that \(\Vert U_{a_0}(t_m, s) u_m - U_{a_0}(t_m, s) u_0\Vert _{L_p(D)} < \epsilon /2\) for \(m \ge m_2\). Consequently,
for \(m \ge \max \{m_1, m_2\}\). \(\square \)
3.1 The adjoint operator
For a fixed \(0<s\le T\) together with ((\(\widehat{\textrm{ME}}\)))\(_{a_0}\)+ ((\(\widehat{\textrm{BC}}\)))\(_{a_0}\) we consider the adjoint equations, that is the backward parabolic equations
complemented with the boundary conditions:
where \({\mathcal {B}}^{*}_{a_0}u = {\mathcal {B}}_{a_0^*}u\) with \(a_0^* := ((a_{ji})_{i,j=1}^N,-(b_i)_{i=1}^N,-(a_i)_{i=1}^N, c_0,d_0)\) and \({\mathcal {B}}_{a_0^*}\) is as in (\(\widehat{\textrm{BC}}\))\(_{a_0}\) with \(a_0\) replaced by \(a_0^*\).
Since all analogs of the assumptions (DA1) and (DA2) are satisfied for (2.6)+(2.7), we can define, for \(u_0 \in L_2(D)\), a global (weak) solution of (2.6)\(_{a^*_0}\)+(2.7)\(_{a^*_0}\), defined on [0, s], with the final condition \(u(s) = u_0\). The following analog of Proposition 2.2 holds.
Proposition 2.10
For \(a_0 \in Y_0\), \(0<s\le T\) and \(u_0 \in L_2(D)\) there is precisely one global weak solution
of (2.6)\(_{a^*_0}\)+ (2.7)\(_{a^*_0}\) satisfying the final condition \(u^{*}(s; a_0, u_0) = u_0\). This mapping has the following properties
From now on, s and t will play a role as in the ((\(\widehat{\textrm{ME}}\)))\(_{a_0}\)+ ((\(\widehat{\textrm{BC}}\)))\(_{a_0}\).
Below we formulate an analog of Proposition 2.3.
Proposition 2.11
-
(i)
Let \(1 \le p < \infty \) and \((s,t)\in \dot{\varDelta }\). Then \(U^{*}_{a_0}(s,t)\) extends to a linear operator in \({\mathcal {L}}(L_{p}(D)).\)
-
(ii)
Let \(1< p < \infty \), \(0<s\le T\) and \(a_{0} \in Y_0\). Then, the mapping
$$\begin{aligned} \bigl [\, [0,s] \ni t \mapsto U^{*}_{a_{0}}(s,t) \in {\mathcal {L}}_{\textrm{s}}(L_{p}(D)) \,\bigr ] \end{aligned}$$is continuous.
The following analog of Proposition 2.7(i) holds.
Proposition 2.12
For any \(a_{0} \in Y_0\), any \(0\le t<s\le T\) and any \(1 \le p \le q \le \infty \) there holds \(U^{*}_{a_{0}}(s,t) \in {\mathcal {L}}_{\textrm{s}}(L_p(D), L_q(D))\).
Proposition 2.13
For \(a_0 \in Y_0\) there holds
Proposition 2.13 states that the linear operator \(U^{*}_{a_0}(s, t) \in {\mathcal {L}}(L_2(D))\) is the dual (in the functional-analytic sense) of \(U_{a_0}(t,s) \in {\mathcal {L}}(L_2(D))\). For a proof, see [35, Prop. 2.3.3].
Proposition 2.14
For \(1< p < \infty \) and \(a_0 \in Y_0\) there holds
for any \((s,t)\in \dot{\varDelta }\), \(u_0 \in L_p(D)\) and \(v_0 \in L_{p'}(D)\).
Proof
Fix \((s,t)\in \dot{\varDelta }\), \(u_0 \in L_p(D)\) and \(v_0 \in L_{q'}(D)\). From Propositions 2.7(i) and 2.12 it follows that \(U_{a_0}(\zeta ,s) u_0, U^{*}_{a_0}(\zeta ,t) v_0 \in L_2(D)\) for all \(\zeta \in (s, t)\), consequently \(\langle U_{a_0}(\zeta ,s) u_0, U_{a_0}(\zeta ,t) v_0 \rangle _{L_2(D)}\) is well defined for such \(\zeta \). An application of (2.2), Proposition 2.13 and (2.9) gives that for any \(s< \zeta _1 \le \zeta _2 < t\) there holds
Therefore, the assignment
is constant (denote its value by A). If we let \(\zeta \nearrow t\), then \(U_{a_0}(\zeta ,s) u_0\) converges, by Proposition 2.4, in the \(L_p(D)\)-norm to \(U_{a_0}(t,s) u_0\) and \(U^{*}_{a_0}(\zeta ,t) v_0\) converges, by Proposition 2.11(ii), in the \(L_{p'}(D)\)-norm to \(v_0\), consequently \(\langle U_{a_0}(t,s) u_0, v_0 \rangle _{L_p(D), L_{p'}(D)} = A\). If we let \(\zeta \searrow s\), then \(U_{a_0}(\zeta ,s) u_0\) converges, by Proposition 2.3(ii), in the \(L_p(D)\)-norm to \(u_0\) and \(U^{*}_{a_0}(\zeta ,t) v_0\) converges, by Propositions 2.12 and 2.11(ii), in the \(L_{p'}(D)\)-norm to \(U^{*}_{a_0}(s,t) v_0\), consequently \(\langle u_0, U^{*}_{a_0}(s,t) v_0 \rangle _{L_p(D), L_{p'}(D)} = A\). This concludes the proof. \(\square \)
It follows from Proposition 2.14 that the linear operator \(U^{*}_{a_0}(s, t) \in {\mathcal {L}}(L_{p'}(D))\) is the dual (in the functional-analytic sense) of \(U_{a_0}(t,s) \in {\mathcal {L}}(L_p(D))\).
In the light of the above, the following counterpart to Proposition 2.7(ii) holds.
Proposition 2.15
There are constants \(M \ge 1\) and \(\gamma \in \mathbb {R}\), the same as in Proposition 2.7, such that
for \(1 \le p \le q \le \infty \), \(a_{0} \in Y_{0}\) and \((s,t)\in \dot{\varDelta }\).
3.2 Continuous dependence of weak solutions
Lemma 2.1
Let \(1< p < \infty \) and \(a_0 \in Y_0\). Then, the mapping
is continuous.
Proof
If \(t_m \rightarrow t > s\) as \(m \rightarrow \infty \), \(U_{a_0}(t_m, s)u_0 \rightarrow U_{a_{0}}(t, s)u_0\) in \(L_p(D)\), by Proposition 2.4.
Assume that \(s_m \rightarrow s < t\) as \(m \rightarrow \infty \). Fix \(u_0 \in L_p(D)\) and \(v_0 \in L_{p'}(D)\). We have, by Proposition 2.14 and the adjoint equation analog of Proposition 2.4,
so \(U_{a_{0}}(t, s_m)u_0 \rightharpoonup U_{a_{0}}(t, s)u_0\) in \(L_p(D)\). As \(\{\, U_{a_{0}}(t, s_m)u_0 : m \in \mathbb {N}\,\}\) is, by Proposition 2.9, precompact in \(L_p(D)\), the convergence is in the norm.
Finally, assume that \(s_m \rightarrow s\) and \(t_m \rightarrow t\) with \(s < t\), and fix \(u_0 \in L_p(D)\). We can assume that \(s_m< (s + t)/2 < t\) for all m. By the previous paragraph, \(U_{a_{0}}( (s + t)/2, s_m)u_0 \rightarrow U_{a_{0}}((s + t)/2, s)u_0\) in \(L_p(D)\). Corollary 2.1 implies that
where the convergence is in \(L_p(D)\), too. \(\square \)
Proposition 2.16
Let \(1< p < \infty \) and \(a_0\in Y_0\). Then, the mapping
is continuous.
Proof
Let \(s_m \rightarrow s\) and \(t_m \rightarrow t\) with \(s < t\). Suppose to the contrary that there are \(\epsilon > 0\) and \((u_m)_{m = 1}^{\infty } \subset L_p(D)\), \(\Vert u_m\Vert _{L_p(D)} = 1\), such that
It follows from Proposition 2.9 that, after possibly taking a subsequence and relabeling, we can assume that \(U_{a_0}(t_m, s_m) u_m\) converge to \({\tilde{u}}\) and \(U_{a_0}(t, s) u_m\) converge to \({\hat{u}}\), both in \(L_p(D)\). For any \(v_0 \in L_{p'}(D)\) we have, by Proposition 2.14,
Since \(\Vert u_m\Vert _{L_p(D)} = 1\), we conclude from the adjoint equation analog of Lemma 2.1 that the above expression converges to zero as \(m \rightarrow \infty \). Consequently, \({\tilde{u}} = {\hat{u}}\), a contradiction. \(\square \)
Proposition 2.17
Assume, in addition, (DA4). For \(1< p < \infty \) the mapping
is continuous.
Proof
It follows from [35, Props. 2.2.12 and 2.2.13] that, for \(2 \le p < \infty \), the mapping
is continuous, too.
To conclude the proof it suffices to show that for any \(1< p < 2\) the mapping
is continuous. Observe that if we have \(a_{0,m} \rightarrow a_0 \in Y_0\), \(s_m \rightarrow s\), \(t_m \rightarrow t\) with \(s_m < t_n\) and \(s < t\), and \(u_{0,m} \rightarrow u_0 \in L_p(D)\), then from Proposition 2.9 it follows that, after possibly choosing a subsequence, there is \(w \in L_2(D)\) such that \(U_{a_{0,m}}(t_m, s_m) u_{0,m} \rightarrow w\) in \(L_2(D)\). Consequently, \(\langle U_{a_{0,m}}(t_m, s_m) u_{0,m}, v \rangle _{L_2(D)} \rightarrow \langle w, v \rangle _{L_2(D)}\) as \(m \rightarrow \infty \), for any \(v \in L_2(D)\). On the other hand, one has, by Proposition 2.14,
As \(2< p' < \infty \), an application of the result already obtained to the adjoint equation yields that \(U^{*}_{a_{0,m}}(s_m,t_m) v\) converges, as \(m \rightarrow \infty \), to \(U^{*}_{a_0}(s,t) v\) in \(L_{p'}(D)\). As \(u_{0,m}\) converges to \(u_0\) in \(L_p(D)\), we have that \(\langle u_{0,m}, U^{*}_{a_{0,m}}(s_m,t_m) v \rangle _{L_p(D), L_{p'}(D)}\) converges to \(\langle u_0, U^{*}_{a_0}(s,t) v \rangle _{L_p(D), L_{p'}(D)}\), which is, by Proposition 2.14, equal to \(\langle U_{a_0}(t,s) u_0, v \rangle _{L_2(D)}\). As \(v \in L_2(D)\) is arbitrary, we have \(w = U_{a_0}(t,s) u_0\). \(\square \)
Proposition 2.18
Assume, in addition, (DA4). For \(1< p < \infty \) the mapping
is continuous.
Proof
In order not to overburden the notation, we assume \(s = 0\).
Let \((a_{0, m})_{m = 1}^{\infty } \subset Y_0\) be a sequence converging to \(a_0\) as \(m \rightarrow \infty \), and let \((t_m)_{m = 1}^{\infty } \subset (0,T]\) be a sequence converging to \(t > 0\) as \(m \rightarrow \infty \). Suppose to the contrary that \(\Vert U_{a_{0,m}}(t_m, 0) -U_{a_{0}}(t, 0)\Vert _{{\mathcal {L}}(L_p(D))}\) does not converge to 0, that is, there exist \(\epsilon > 0\) and a sequence \((u_m)_{m = 1}^{\infty } \subset L_p(D)\), \(\Vert u_m\Vert _{L_p(D)} = 1\) for all m, such that
for all m.
It follows from Proposition 2.9 that, after possibly extracting a subsequence, we can assume that \(U_{a_{0,m}}(t_m/2, 0) u_m\) and \(U_{a_{0}}(t/2, 0) u_m\) converge, as \(m \rightarrow \infty \), in the \(L_p(D)\)-norm. We claim that both converge to the same \({\tilde{u}}\). Indeed, it suffices to check that the difference \((U_{a_{0,m}}(t_m/2, 0) -U_{a_{0}}(t/2, 0)) u_m\) converges to zero in \(L_p(D)\), which is, in light of the equalities
a consequence of the analog for the adjoint equation of Proposition 2.17.
Proposition 2.17 implies that
and
therefore \(\Vert U_{a_{0,m}}(t_m, 0) u_m -U_{a_{0}}(t, 0) u_m\Vert _{L_p(D)}\), converges to zero, a contradiction.
\(\square \)
4 Mild solutions
In the present section, we assume (DA1), (DA2) and that Y as in Definition 1.1 is such that its flattening \(Y_0\) satisfies (DA3). Occasionally, we will assume (DA5).
Definition 3.1
(Multiplication Operator) For \(a \in Y\), \(1 \le p\le \infty \) and \(0\le t\le T\) we define multiplication operator \(C^{1}_{a}(t):L_p(D)\rightarrow L_p(D)\) as follows
The \(C^{1}_{a}(t)\) operator is well defined as long as assumption (DA2) holds. To be more precise, we use a corollary from assumption (DA2) on t-sections of \(c_1\).
Lemma 3.1
(Boundedness of Multiplication Operator) The multiplication operator \(C_{a}^1(t)\) is linear and bounded uniformly with respect to a.e. \(0< t< T \) and \(a\in Y\).
It should be remarked that the exceptional sets can be different for different \(a \in Y\).
Proof
Let K be the norm bound of Y (see assumption (Y1)). For any \(v\in L_p(D)\) by the Hölder inequality we get
where above inequality holds for a.e. \(0< t< T\), so the operator norm of \(C^{1}_{a}(t)\) is bounded a.e. by \(\Vert c_1(t,\cdot )\Vert _{L_{\infty }(D)}\) what can be bounded uniformly with respect to a.e. \(0< t< T\) by virtue of assumption (DA2). Since Y is bounded by K we also have uniform boundedness in \(a \in Y\). \(\square \)
Below we present a series of lemmas to prove the measurability of individual parts of the mild solution. We will make frequent use of Lemma 0.1 in this part of the work, in particular, Remark 0.1 on measurability will also be useful.
Lemma 3.2
For any \(1< p < \infty \) and any norm bounded set \(E \subset C([-1,T],L_p(D))\) the set
is bounded in \(L_{\infty }((0,T),L_p(D))\).
Proof
Let \(u\in E\) and \(\Phi \in \widetilde{{\mathcal {R}}}\). The mapping \(u \, \circ \, \Phi \) is \(({\mathfrak {L}}((0,T)), {\mathfrak {B}}(L_p(D)))\)-measurable, since for any fixed open set \(V\subset L_p(D)\) the preimage \(u^{-1}[V]\) is open, hence \((u\circ \Phi )^{-1}[V]\in {\mathfrak {L}}((0,T))\). Moreover, the \(L_{\infty }((0,T), L_p(D))\)-norm of the map \(u\circ \Phi \) is uniformly bounded with respect to u and \(\Phi \) by the same constant as E due to the inequality
\(\square \)
Lemma 3.3
For any \(1< p < \infty \) and any norm bounded set \({\tilde{E}} \subset L_{\infty }((0,T),L_p(D))\) the set
is bounded in \(L_{\infty }((0,T),L_p(D))\).
Proof
Let \({\tilde{u}}\in {\tilde{E}}\) and \(a\in Y\). From Lemma 0.1(a) follows that the mapping
is \(({\mathfrak {L}}((0, T))\otimes {\mathfrak {L}}(D), {\mathfrak {B}}(\mathbb {R}))\)-measurable. Hence, the function
for any \(a\in Y\) is \(({\mathfrak {L}}((0, T))\otimes {\mathfrak {L}}(D), {\mathfrak {B}}(\mathbb {R}))\)-measurable, since it can be rewritten as the product of \(({\mathfrak {L}}((0, T))\otimes {\mathfrak {L}}(D), {\mathfrak {B}}(\mathbb {R}))\)-measurable functions, namely
It suffices now to notice that for a.e. \(t \in (0,T)\) the t-section of (3.3) belongs (by the definition of the multiplication operator) to \(L_p(D)\). So from Lemma 0.1(b) it follows that the mapping
is \(({\mathfrak {L}}((0, T)), {\mathfrak {B}}(L_p(D)))\)-measurable. By the norm estimate,
we obtain the statement. \(\square \)
Lemma 3.4
Assume \(1< p < \infty \), \(a_0 \in Y_{0}\), and \(u \in L_{\infty }((0, T), L_p(D))\). Then
-
(i)
for any \(0 < t \le T\) the function
$$\begin{aligned}{}[\, \zeta \mapsto U_{a_0}(t, \zeta ) u(\zeta ), \text { for a.e. } \zeta \in (0, t) \,] \end{aligned}$$(3.5)belongs to \(L_{\infty }((0, t), L_{p}(D))\) moreover, the linear operator assigning (3.5) to u belongs to \({\mathcal {L}}(L_{\infty }((0,1), L_p(D)), L_{\infty }((0,1), L_p(D)))\), with the norm bounded uniformly in \(a_0 \in Y_0\),
-
(ii)
the mapping
$$\begin{aligned} \biggl [\, [0, T] \ni t \mapsto \int \nolimits _{0}^{t} U_{a_0}(t, \zeta ) u(\zeta ) \, \textrm{d}\zeta \,\biggr ] \end{aligned}$$(3.6)belongs to \(C([0, T], L_p(D))\).
Proof
Fix \(0 < t \le T\). We show first that (3.5) defines a \(({\mathfrak {L}}((0, t)), {\mathfrak {B}}(L_{p}(D)))\)-measurable function. It is equivalent, by Theorem 0.1, to showing that for each \(v \in L_{p'}(D)\) the function
is \(({\mathfrak {L}}((0, t)), {\mathfrak {B}}(\mathbb {R}))\)-measurable. By Proposition 2.14, for Lebesgue-a.e. \(\zeta \in [0, t)\) there holds
It suffices now to notice that u is \(({\mathfrak {L}}((0, t)), {\mathfrak {B}}(L_p(D)))\)-measurable, by assumption, and that the function
is continuous, by the adjoint equation analog of Lemma 2.1. It follows from Proposition 2.7(ii) that the function
belongs to \(L_{\infty }((0, t))\). Then the membership of (3.5) in \(L_{\infty }((0, t), L_{p}(D))\) as well as the bound on its norm follow from the generalized Hölder inequality. The proof of part (i) is thus completed.
We proceed to the proof of part (ii). By part (i), the function (3.6) is well defined. Let \(0 \le t_1 \le t_2 \le T\). We write
Let \(\epsilon > 0\). As (3.5) belongs to \(L_{\infty }((0, t), L_{p}(D))\), it is a consequence of [16, Thm. II.2.4(i)] that the \(L_p(D)\)-norm of the second term on the right-hand side can be made \(< \epsilon /3\) by taking \(t_1, t_2\) sufficiently close to each other. Regarding the first term, we write
Again by [16, Thm. II.2.4(i)], for \(\eta > 0\) sufficiently small there holds
It follows from Proposition 2.16 that the assignment
is uniformly continuous, consequently there exists \(\delta > 0\) such that if \(\eta \le \zeta + \eta \le t_1 \le t_2\), \(t_2 - t_1 < \delta \), then
Therefore,
This concludes the proof of part (ii). \(\square \)
Definition 3.2
(Mild Solution) For \(1 \le p<\infty \), \(a \in Y\), \(0 \le s < T_0 \le T\) and \(u_0 \in C([s-1,s], L_p(D))\) and \(R \in {\mathcal {R}}\) the function \(u\in C([s-1,T_0], L_p(D))\) such that
holds and the integral equation
is satisfied in \(L_p(D)\) on \([s,T_0]\) will be called a mild solution of (ME)\(_a+\)(BC)\(_{a}\).
For \(T_0 = T\), we have a global mild solution.
At first note that the concept of mild solution, especially part (3.8), is well defined based on Lemma 3.3 and Lemma 3.4. At some moments, we use the name “mild solution” to describe function \(u\upharpoonright _{[0,T_0]}\) instead of \(u\in C([-1,T_0],L_p(D))\) satisfying (3.7) and (3.8). This convention seems more natural especially in the context of continuous dependence on coefficients. A similar convention can be found in the literature [28].
4.1 Existence and uniqueness of global mild solutions
Proposition 3.1
There exists \(\Theta _0 \in (0,T]\) such that for any \(1<p<\infty \), \(a \in Y \), \(R\in {\mathcal {R}}\), \(0\le s\le T-\Theta _0\), \(u_0\in C([s-1,s],L_p(D))\) and any \(0<\Theta \le \Theta _0\) there exist unique solution of (ME)\(_{a}\)+ (BC)\(_{a}\) on \([s-1,s+\Theta ]\) with initial condition \(u_0\).
Proof
The idea of the proof runs as follows. The solution is obtained as a fixed point of the contraction mapping \({\mathfrak {G}}\) of \(C([s,s+\Theta ], L_p(D))\) into itself (see 2.3(ii), (ii) and 3.3) defined as
where \(s \le t \le s+\Theta \) and \(\Theta \in (0, T]\) is sufficiently small. Until revoking, u and v stand for generic functions in \(C([s,s+\Theta ], L_p(D))\). For such a u, we interpret \(u(\zeta - R(\zeta ))\) (similarly v) as \(u_0(\zeta - R(\zeta ))\) when \(\zeta -R(\zeta ) \in (-1, 0)\).
By taking \(\displaystyle 0< \Theta \le \Theta _0 :=1/(2MK e^{\gamma T})\) we obtain that the contraction coefficient is less than 1. \(\square \)
The contraction mapping principle guarantees the existence and uniqueness of the fixed point u of \({\mathfrak {G}}\), which is then the unique mild \(L_p\)-solution of (ME)\(_{a}\)+ (BC)\(_{a}\) on \([s-1, \Theta _0]\) satisfying the initial condition (IC).
Lemma 3.5
For any \(1<p<\infty \), \(0<s_1<s_2\le T\), \(a\in Y\), \(R\in {\mathcal {R}}\), \(u_0\in C([-1,0],L_p(D))\) and \(v :[-1,s_2]\rightarrow L_p(D)\) the following statements are equivalent:
-
(i)
a function v is the mild solution of (ME)\(_{a}\)+ (BC)\(_{a}\) on \([-1,s_2]\) with initial condition \(u_0\),
-
(ii)
a function \(v\!\!\restriction _{[-1,s_1]}\) is the mild solution of (ME)\(_{a}\)+ (BC)\(_{a}\) with initial condition \(u_0\) and \(v\!\!\restriction _{[s_1-1,s_2]}\) is the mild solution of (ME)\(_{a}\)+ (BC)\(_{a}\) with initial condition \(v\!\!\restriction _{[s_1-1,s_1]}\).
Proof
Let \(p,s_1,s_2,a,R,u_0\) and v be as in the statement. To prove \((i) \Rightarrow (ii)\) it suffices to see that for any \(s_1 \le t\le s_2\), in view of (2.2) and [1, Lemma 11.45] there holds
In order to prove \((ii)\Rightarrow (i)\) fix \(t\in [-1,s_2]\) and consider the cases: \(t\in [-1,0]\), \(t\in [0,s_1]\) or \(t\in [s_1,s_2]\). Since the first two cases are straightforward and the third is a similar calculation to (3.10), the proof is finished. \(\square \)
Theorem 3.1
For any \(1< p < \infty \), \(a \in Y\), \(u_0 \in C([-1,0],L_p(D))\) and \(R \in {\mathcal {R}}\) equation (ME)\(_{a}\)+ (BC)\(_{a}\) has a unique global mild solution on [0, T].
Proof
Fix a, \(u_0\) and R as in the statement. Let
It suffices to prove that \(T\in Q\). Suppose to the contrary that \(T\not \in Q\). Since \(\Theta _0 \in Q\) (where \(\Theta _0\) stands for constant obtained in Proposition 3.1) and \(Q \subset [0,T]\), \(\sup Q < \infty \). It is straightforward that \(0 \le \sup Q-\Theta _0/2<\sup Q\le T\) hence there exists \(s \in Q\) such that \(s > \sup Q - \Theta _0/2\).
Let \(v_1 :[-1,s] \rightarrow L_p(D)\) be the unique mild solution with initial condition \(u_0\). From the definition of \(\Theta \) it follows that there is a mild solution \(v_2 :[s-1,\min \{s+\Theta _0,T\}] \rightarrow L_p(D)\) with initial condition \(v_1\!\!\restriction _{[s-1,s]}\). Let
We claim that v is a unique mild solution of (ME)\(_{a}\)+ (BC)\(_{a}\) on \([-1,\min \{s+\Theta _0,T\}]\) with initial condition \(u_0\). From Lemma 3.5 it follows that v is in fact a mild solution. For uniqueness, assume \(w :[-1, \min \{ s+\Theta _0, T \}]\rightarrow L_p(D)\) is any mild solution. Then clearly \(w\!\!\restriction _{[-1, s]} = v_1\). Moreover, by Lemma 3.5, the function \(w\!\!\restriction _{[s-1, \min \{ s+\Theta _0, T \}]}\) is a mild solution with initial condition \(w\!\!\restriction _{[s-1, s]} = v_1\!\!\restriction _{[s-1, s]}\), so by the uniqueness of \(v_2\) we have that \(w\!\!\restriction _{[s-1, \min \{ s+\Theta _0, T \}]} = v_2\). Hence \(v = w\). The proof is completed by the following observation: if \(\min \{s+\Theta _0,T\}=s+\Theta _0\) then \(s+\Theta _0 \in Q\), so we get a contradiction with the fact that \(s+\Theta _0 > \sup Q\): otherwise \(T\in Q\), which contradicts the assumption. \(\square \)
The above result allows us to define a mild solution of (ME)\(_{a}\)+ (BC)\(_{a}\) on the whole of \([-1, T]\) or \([s-1,T]\) if necessary.
For \(s = 0\), to stress the dependence of the solution on a, \(u_0\), R we write \(u(\cdot ; a, u_0,R)\). For \(t \in [-1, 0]\), \(u(t; a, u_0, R)\) is interpreted as \(u_0(t)\). Moreover, when it does not lead to confusion, we sometimes write \(u(t;a, u_0, \Phi )\) instead of \(u(t; a, u_0, R)\).
4.2 Compactness of solution operator
Lemma 3.6
Assume \(1< p< \infty \) and \(0 < T_1 \le T\). Then for any bounded \(F \subset L_{\infty }((0, T), L_p(D))\) the set
is precompact in \(L_p(D)\).
Proof
Compare [35, Thm. 6.1.3]. Fix p, \(T_1\) and F as in the statement. Let \((t_{m})_{m=1}^{\infty } \subset [T_1,T]\), \((a_{0,m})_{m=1}^{\infty } \subset Y_{0}\), \((u_{m})_{m=1}^{\infty } \subset F\). We claim that for any fixed \(l \in \mathbb {N}\) the set
is precompact in \(L_{p}(D)\). Denote by \(M_0 > 0\) the supremum of the \(L_{\infty }((0, T), L_{p}(D))\)-norms of \(u_m\), and put 
to be the closure in \(L_{p}(D)\) of the set
The set 
is balanced. We have 
. By Proposition 2.9, 
is compact. [16, Cor. II.2.8] implies that
where \({{\,\mathrm{\overline{co}}\,}}\) denotes the closed convex hull in \(L_p(D)\). As, by Mazur’s theorem ( [16, Thm. II.2.12]), 
is compact for any \(l \in \mathbb {N}\), this proves our claim that \({\widetilde{F}}_{l}\) are precompact in \(L_{p}(D)\). By a diagonal process we can assume without loss of generality that for each \(l \in \mathbb {N}\) the integrals
converge, as \(m \rightarrow \infty \), in \(L_{p}(D)\).
Lemma (i) guarantees that the functions \(U_{a_{0,m}}(t_m, \cdot ) u_m(\cdot )\) belong to \(L_{\infty }((0, t_m), L_p(D))\), with their \(L_{\infty }((0, t_m), L_p(D))\)-norms bounded uniformly in m. We estimate, via Hölder’s inequality,
It follows from Proposition 2.7(ii) that for any \(\epsilon > 0\) there is \(l_0 \in \mathbb {N}\) such that
uniformly in \(m \in \mathbb {N}\). By the previous paragraph, there is \(m_0\) such that if \(m_1, m_2 \ge m_0\) then
Therefore,
for any \(m_1, m_2 \ge m_0\).
From this, it follows that
is a Cauchy sequence in \(L_{p}(D)\). Therefore, \({\widehat{F}}\) is precompact in \(L_{p}(D)\). \(\square \)
Lemma 3.7
For any \(1< p < \infty \) and any bounded \(F \subset L_{\infty }((0, T), L_p(D))\) the set
is precompact in \(C([0, T], L_{p}(D))\).
Proof
By the Ascoli–Arzelà theorem, it suffices, taking Lemma 3.6 into account, to show that for any \(\epsilon > 0\) there is \(\delta > 0\) such that, if \(0 \le t_1 \le t_2 \le T\), \(t_2 - t_1 < \delta \), then
for all \(a_0 \in Y_{0}\) and all \(u \in F\). In order not to introduce too many constants we assume that F equals the unit ball in \(L_{\infty }((0, T), L_p(D))\).
We write
By Proposition 2.7(ii),
provided \(t_2 - t_1 < \epsilon /(3 M e^{\gamma T})\).
Further, we write
By Proposition 2.7(ii), if \(0< \eta < \epsilon /(6M e^{\gamma T})\), then
It follows from Proposition 2.18 that the assignment
is uniformly continuous, consequently there exists \(\delta > 0\) such that if \(\eta \le s_1 < s_2 \), \(s_2 - s_1 < \delta \), then
Therefore,
The estimates (3.11), (3.12) and (3.2) do not depend on the choice of \(a_0 \in Y_0\), so gathering them gives the required property. \(\square \)
Theorem 3.2
For any \(0 < T_1 \le T\), any \(1< p < \infty \) and any bounded \(E \subset C([- 1, 0], L_p(D))\) the set
is precompact in \(C([T_1, T], L_p(D))\).
Proof
We will use the notation \(I_i(t; a, u_0,R)\), \(i = 0, 1\) where
taking account of the parameter a and the initial value \(u_0\). The precompactness of the set
in \(C([T_1, T], L_p(D))\) is a consequence of Proposition 2.9.
In order to prove the precompactness in \(C([T_1, T], L_p(D))\) of
it suffices to use results from Lemma 3.3 and Lemma 3.7. \(\square \)
Theorem 3.2 leads to the following conclusion about precompactness of the solutions up to zero. Since under additional assumption (DA5) for a fixed \(u_0 \in C([-1,0],L_p(D))\) the set
is simply a singleton, this observation combined with Lemma 3.7 leads to the following result.
Theorem 3.3
Assume additionally (DA5). For any \(1< p < \infty \) and any \(u_0 \in C([- 1, 0], L_p(D))\) the set
is precompact in \(C([0, T], L_p(D))\).
5 Continuous dependence on initial conditions
In the present section, we assume (DA1), (DA2) and that Y as in Definition 1.1 is such that its flattening \(Y_0\) satisfies (DA3). Further, \(1< p < \infty \).
Definition 4.1
For \(t\in [0,T]\), \(a \in Y\), \(u_0 \in C([-1,0],L_p(D))\) and \(R \in {\mathcal {R}}\) we define
For notational simplicity, we often write \(u(t+\,\cdot )\) instead of \(u(t+\, \cdot \, ;a,u_0,R)\) and \(\delta (t)\) instead of \(\delta (t;a,u_0,R)\) when \(a \in Y\) and \(u_0 \in C([-1,0],L_p(D))\) are fixed and this does not lead to confusion.
Lemma 4.1
For any \(a \in Y\), \(u_0 \in C([-1,0],L_p(D))\) and \(R \in {\mathcal {R}}\) the function \(\delta (\cdot ;a,u_0,R) :[0,T] \rightarrow \mathbb {R}^+\) is continuous.
Proof
First note that the mapping
is continuous as a composition of continuous mappings. Due to the compactness of \([-1,0]\), the \(\delta (\cdot ,a,u_0,R)\) function is continuous when \(a \in Y\), \(R\in {\mathcal {R}}\) and \(u_0\in C([-1,0],L_p(D))\) are fixed. \(\square \)
Lemma 4.2
There are constants \(M_1, M_2\) such that for any \(\rho \in [0,T]\) the inequality
holds for all \(a \in Y\), \(u_0\in C([-1,0],L_p(D))\) and \(R \in {\mathcal {R}}\).
Proof
Fix \(\rho \in [0,T]\) and note that
where M is a uniform bound of the operator \(U_{{\tilde{a}}}(t)\) with respect to \(a \in Y\) and \(0 \le t \le T\) (see Proposition 2.7(ii)), the constant K is a uniform bound of the operator \(C^{1}_{a}(\zeta )\) with respect to \(a \in Y\) and \(0 \le t \le T\) (see Lemma 3.1). Moreover, the bounds M and K are independent on initial condition \(u_0\). By setting \(M_1 = Me^{\gamma T}\), \(M_2 = MKe^{\gamma T}\), we end the proof. \(\square \)
From now on, throughout this section the constants \(M_1\) and \(M_2\) will be defined as in Lemma 4.2.
Proposition 4.1
For any sequence \((u_{0,m})_{m=1}^{\infty }\subset C([-1,0],L_p(D))\) convergent to zero, any \(t \in [0,T]\), \(R\in {\mathcal {R}}\) and \(a \in Y\) the sequence \(\delta _m(t) := \delta (t;a,u_{0,m},R)\) converges to zero.
Proof
Fix \(t \in [0,T]\) and \(-1\le \vartheta \le 0\) and let us consider two cases.
-
If \(0\le t+\vartheta \le T\), then from Lemma 4.2 there holds
$$\begin{aligned} \Vert u(t+\vartheta )\Vert _{L_p(D)}&\le M_1 \delta (0)+ M_2\int \nolimits _{0}^{t+\vartheta } \delta (\zeta )\,\textrm{d}\zeta \\&\le M_1 \delta (0)+ M_2\int \nolimits _{0}^{t} \delta (\zeta )\,\textrm{d}\zeta . \end{aligned}$$ -
If \(-1\le t+\vartheta \le 0\), then the inequality
$$\begin{aligned} \Vert u(t+\vartheta )\Vert _{L_p(D)} \le M_1 \delta (0)+ M_2\int \nolimits _{0}^{t} \delta (\zeta )\,\textrm{d}\zeta \end{aligned}$$is straightforward, as even the stronger one \(\Vert u(t+\vartheta )\Vert _{L_p(D)} \le M_1 \delta (0)\) is true.
Applying \(\sup \) with respect to \(\vartheta \) on both sides give us that
what can be rewritten in terms of the \(\delta \) function as
The function \(\delta \) is nonnegative and continuous on the compact domain, hence it is integrable. Using the Grönwall lemma we get
\(\square \)
Lemma 4.2 and Proposition 4.1 lead to global \(L_p\)-norm estimation of the mild solution of \((ME)_a+(BC)_a\) in terms of initial conditions.
Proposition 4.2
There is constant \(\overline{M} > 0\) such that inequality
holds for any \(1<p<\infty \), \(t\in [0,T]\), \(a\in Y\), \(R\in {\mathcal {R}}\) and \(u_0 \in C([-1,0],L_p(D))\).
Proof
Let \(\overline{M} = M_1\exp (M_2T)\), where \(M_1,M_2\) are constants as in Lemma 4.2. Then, by Proposition 4.1 we can write
\(\square \)
Proposition 4.3
For any \(a\in Y\) and \(R\in {\mathcal {R}}\) the mapping
is continuous.
Proof
Let \(a\in Y\), \(R\in {\mathcal {R}}\) be fixed. Then, in the spirit of Cauchy’s definition we can find that
for any initial conditions \(u_{0,1}, u_{0,2}\in C([-1,0],L_p(D))\). The first inequality results from the linearity of the problem \((ME)_{a}+(BC)_a\) and the second inequality follows from (4.1). \(\square \)
6 Continuous dependence on coefficients and delay
In the present section, we assume (DA1), (DA2) and that Y as in Definition 1.1 is such that its flattening \(Y_0\) satisfies (DA3) and (DA4). As in Sect. 4, \(1< p < \infty \).
Proposition 5.1
For any \(0 < T_1 \le T \), \(R \in {\mathcal {R}}\) and \(u_0 \in C([- 1, 0], L_p(D))\) the mapping
is continuous.
Proof
Fix p, \(T_1\), R and \(u_0\) as in the proposition. Let \((a_{m})_{m=1}^{\infty } \subset Y\) converge to a. Put \(u_m(\cdot )\) for \(u(\cdot ; a_m, u_0,R)\) and \(u(\cdot )\) for \(u(\cdot ; a, u_0,R)\). It suffices to prove that there is a subsequence \((a_{m_k})_{k=1}^{\infty } \subset Y\) such that \(u_{m_k}(\cdot )\) converges to \(u(\cdot )\) on \([T_1,T]\) uniformly. By Theorem 3.2 via diagonal process, we can find a subsequence \(u_{m_k}\) such that \(u_{m_k}\!\!\restriction _{(0, T]}\) converge to some continuous \({\hat{u}} :(0, T] \rightarrow L_{p}(D)\) and the convergence is uniform on compact subsets of (0, T]. The function \({\hat{u}}\) is clearly bounded by Proposition 4.2. Moreover, we extend the map \({\hat{u}}\) to the whole \([-1,T]\) by \(u_0\) on \([-1,0]\), i.e., now
It remains to prove that \({\tilde{u}} = u\). In order not to overburden the notation, we write \(u_m\) instead of \(u_{m_k}\).
Our first step is to show that, for each \(t \in (0, T]\),
in the \(L_{p}(D)\)-norm as \(m \rightarrow \infty \). The convergence in (5.1) is a consequence of Proposition 2.17. The convergence in (5.2) can be shown by showing the convergence of the difference
to zero. Write \(J^{(i)}_{m}(t)\), \(i = 1, 2, 3\), for the i-th term on the right-hand side of (5.3). The convergence of \(J^{(1)}_{m}(t)\) follows from the Lebesgue dominated convergence theorem for Bochner integral: since the integrand
is \(({\mathfrak {L}}((0, t)), {\mathfrak {B}}(L_p(D)))\)-measurable for all \(m\in \mathbb {N}\) (see Lemmas (i) and 3.3) and bounded uniformly (see Proposition 4.2) in \(m\in \mathbb {N}\):
it suffices to check that for a.e. \(\zeta \in (0,t)\) the integrand converges to zero, which follows from the estimate
and Proposition 2.18.
In order to prove \(J_m^{(2)}(t)\rightarrow 0\) as \(m\rightarrow \infty \), we proceed similarly. We see that the mapping
is \(({\mathfrak {L}}((0, t)), {\mathfrak {B}}(L_p(D)))\)-measurable for all \(m \in \mathbb {N}\), as a consequence of Lemmas (i) and 3.3, and bounded uniformly in \(m\in \mathbb {N}\), since, by Proposition 2.7(ii), Lemma 3.1 and Proposition 4.2,
Further, the convergence, for a.e. \(\zeta \in (0,t)\),
in \(L_p(D)\) is due to the pointwise convergence of \(u_m\) to \({\tilde{u}}\) on \([-1,T]\) and the estimate (by Proposition 2.7(ii) and Lemma 3.1)
The convergence of \(J^{(3)}_{m}(t)\) follows from the facts that the set
is precompact in \(L_{p}(D)\) (see Lemmas 3.6, 3.3, Proposition 4.2) and that \(J^{(3)}_{m}(t)\) converge weakly to zero, i.e.,
for any \(v \in L_{{p'}}(D)\), where \(\langle \cdot , \cdot \rangle \) stands for the duality pairing between \(L_{p}(D)\) and \(L_{p'}(D)\). By the Hille theorem ( [16, Thm. II.2.6]) and Proposition 2.14,
Now we need to use a subtler approach based on the convergence \(c_{1,m}\) to \(c_{1}\) in the weak-* topology of \(L_{\infty }((0,t) \times D)\). First note that the mappings
belong to \(L_{\infty }((0,t),L_p(D))\) and \(L_{\infty }((0,t),L_{p'}(D))\), respectively. Therefore the mapping (the product of the above maps)
belong to \(L_{1}((0,t)\times D)\) see Lemma 0.1(a). It suffices now to note that from Fubini’s theorem we have
so the integral tends to zero as \(m\rightarrow \infty \).
We have thus proved that
Now we prove the continuity of the extension \({\tilde{u}}\). Note that the only point where it can fail is \(t=0\). However, this is not the case since the mappings
are continuous (see Lemmas 2.3(ii) and (ii), so the mapping \({\tilde{u}}\) is continuous on the whole \([-1,T]\). Also, \({\tilde{u}} = u_0\) on \([-1,0]\), hence \({\tilde{u}}\) is in fact the mild solution of \((ME)_{{\tilde{a}}}+(BC)_{{\tilde{a}}}\), therefore, by uniqueness, \({\tilde{u}}(t) = u(t;{\tilde{a}},u_0,R)\) for any \(t \in [-1, T]\).
\(\square \)
Proposition 5.2
Assume additionally (DA5). For any \(u_0 \in C([- 1, 0], L_p(D))\) and \({\mathcal {R}}_0\subset {\mathcal {R}}\) satisfying the assumption (DA6) the mapping
is continuous.
Proof
(Sketch of proof) Fix p, \(u_0\) and \({\mathcal {R}}_0\). Let \((a_{m})_{m=1}^{\infty } \subset Y\) converge to a and \((R_m)_{m=1}^{\infty }\subset {\mathcal {R}}_0\) converge to R, and put \(u_m(\cdot )\) for \(u(\cdot ; a_m, u_0,R_m)\) and \(u(\cdot )\) for \(u(\cdot ; a, u_0,R)\). We will proceed as in Proposition 5.1. In particular, \({\tilde{u}}\) has the same meaning. However, we have in fact more: as we assume (DA5), we can apply Theorem 3.3 to show that \(u_m\) converge to \({\hat{u}}\) uniformly on [0, T], from which it follows in particular that \({\tilde{u}}\) is continuous on the whole of \([-1, T]\).
It follows again from (DA5) that \({\tilde{a}}_m = {\tilde{a}}\) for all \(m \in \mathbb {N}\), so
holds trivially. We start by showing that, for each \(t \in [0, T]\),
in the \(L_{p}(D)\)-norm as \(m \rightarrow \infty \). The above convergence can be proved by showing convergence of the terms
to zero. Write \(K^{(i)}_{m}(t)\), \(i = 1, 2\), for the i-th term on the right-hand side of (5.6).
Regarding the convergence of \(K^{(1)}_{m}(t)\) to zero, we show the \(({\mathfrak {L}}((0, t)), {\mathfrak {B}}(L_p(D)))\)-measurability, for all \(m \in \mathbb {N}\), of the integrand
in the same way as in the proof of the convergence of \(J^{(2)}_{m}(t)\) in Proposition 5.1. The fact that for a.e. \(\zeta \in (0,T)\) we have that
in \(L_p(D)\) is due, in view of (DA6), to the uniform convergence of \(u_m\) to \({\tilde{u}}\) on \([-1, T]\) together with the estimate
The proof of the convergence of \(K^{(2)}_{m}(t)\) to zero is just a copy, word for word, of the proof of the convergence of \(J^{(3)}_{m}(t)\) in Proposition 5.1. \(\square \)
Theorem 5.1
-
(i)
For any \(0 < T_1 \le T\) and \(R \in {\mathcal {R}}\) the mapping
$$\begin{aligned}{} & {} \Bigl [\, Y \times C([- 1, 0], L_p(D)) \ni (a, u_0) \\{} & {} \quad \mapsto u(\cdot ; a, u_0,R)\!\!\restriction _{[T_1, T]} \in C([T_1, T], L_{p}(D)) \,\Bigr ] \end{aligned}$$is continuous.
-
(ii)
Under (DA5), if \({\mathcal {R}}_0\subset {\mathcal {R}}\) is a subset such that the assumption (DA6) holds then the mapping
$$\begin{aligned}{} & {} \Bigl [\, Y \times C([- 1, 0], L_p(D))\times {\mathcal {R}}_0 \ni (a, u_0, R) \\{} & {} \quad \mapsto u(\cdot ; a, u_0,R)\!\!\restriction _{[0, T]} \in C([0, T], L_{p}(D)) \,\Bigr ] \end{aligned}$$is continuous.
Proof
Fix \(1< p < \infty \). We start by proving (i), so fix also \(T_1\), R as in the statement. Let a sequence \((a_m)_{m=1}^{\infty }\subset Y\) converge to \(a \in Y\) and \((u_{0,m})_{m=1}^{\infty }\subset C([-1,0],L_p(D))\) converge to \(u_0\in C([-1,0],L_p(D))\). The main idea of the proof is based on the estimation
Proposition 4.2 implies
Therefore, the first part of the right-hand side of (5.7) converges to zero as \(m \rightarrow \infty \). The second part of (5.7) converges to zero by Proposition 5.1. Item (ii) can be proved similarly. So, assume additionally (DA5) and, instead of fixing the delay \(R\in {\mathcal {R}}\) take a sequence \((R_m)_{m=1}^{\infty } \subset {\mathcal {R}}_0\) convergent to some \(R\in {\mathcal {R}}_0\). By similar estimation,
together with Propositions 4.2 and 5.2 concludes the proof. \(\square \)
Data availability statement
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
C. D. Aliprantis and K. C. Border, “Infinite Dimensional Analysis. A Hitchhiker’s Guide,” third edition, Springer, Berlin, 2006.
H. Amann and J. Escher, “Analysis. III,” translated from the 2001 German original by S. Levy and M. Cargo, Birkhäuser, Basel, 2009.
W. Arendt and A. V. Bukhvalov, Integral representations of resolvents and semigroups, Forum Math. 6 (1994), no. 1, 111–135.
L. Barreira, D. Dragičević and C. Valls, Nonuniform spectrum on Banach spaces, Adv. Math. 321 (2017), 547–591.
L. Barreira and C. Valls, “Hyperbolicity in Delay Equations,” Ser. Appl. Comput. Math., 4, World Scientific, Hackensack, NJ, 2021.
A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl. 264 (2001), no. 1, 1–20.
A. Bátkai and S. Piazzera, A semigroup method for delay equations with relatively bounded operators in the delay term, Semigroup Forum 64 (2002), no. 1, 71–89.
A. Bátkai and S. Piazzera, “Semigroups for Delay Equations,” Res. Notes Math., 10, A K Peters, Wellesley, MA, 2005.
A. Blumenthal, A volume-based approach to the multiplicative ergodic theorem on Banach spaces, Discrete Contin. Dyn. Syst. 36 (2016), no. 5, 2377–2403.
C. Chicone and Y. Latushkin, “Evolution Semigroups in Dynamical Systems and Differential Equations,” Math. Surveys Monogr., 70, American Mathematical Society, Providence, RI, 1999.
S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995), no. 2, 429–477.
S.-N. Chow and H. Leiva, Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1071–1081.
D. Daners, Heat kernel estimates for operators with boundary conditions, Math. Nachr. 217 (2000), 13–41.
Z. Denkowski, S. Migórski and N. S. Papageorgiou, “An Introduction to Nonlinear Analysis: Theory,” Kluwer, Boston, MA, 2003.
R. Dautray and J.-L. Lions, “Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems. I,” with the collaboration of M. Artola, M. Cessenat and H. Lanchon, translated from the French by A. Craig, Springer, Berlin, 1992.
J. Diestel and J. J. Uhl, Jr., “Vector Measures,” with a foreword by B. J. Pettis, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977.
T. S. Doan, Lyapunov exponents for random dynamical systems, Ph. D. dissertation, Technische Universität Dresden, 2009.
T. S. Doan and S. Siegmund, Differential equations with random delay, in: Infinite Dimensional Dynamical Systems, Fields Inst. Commun., 64, Springer, New York, 2013, 279–303.
N. Dunford and J. T. Schwartz, “Linear Operators. I. General Theory,” with the assistance of W. G. Bade and R. G. Bartle, Pure and Applied Mathematics, Vol. 7, Interscience, New York and London, 1958.
W. E. Fitzgibbon, Stability for abstract nonlinear Volterra equations involving finite delay, J. Math. Anal. Appl. 60 (1977), no. 2, 429–434.
W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations 29 (1978), no. 1, 1–14.
G. B. Folland, “Real Analysis. Modern Techniques and Their Applications,” second edition, Pure and Applied Mathematics, Wiley, 1984.
G. Fragnelli, A spectral mapping theorem for semigroups solving PDEs with nonautonomous past, Abstr. Appl. Anal. 2003, no. 16, 933–951.
G. Fragnelli and G. Nickel, Partial functional differential equations with nonautonomous past in\(L^p\)-extitphase spaces, Differential Integral Equations 16 (2003), no. 3, 327–348.
G. Froyland, S. Lloyd, and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst. 33 (2013), no. 9, 3835–3860.
C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1230–1272.
C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Mod. Dyn. 9 (2015), 237-255.
J. K. Hale and S. M. Verduyn Lunel, “Introduction to Functional Differential Equations,” Appl. Math. Sci., 99, Springer, New York, 1993.
O. A. Ladyzhenskaya [O. A. Ladyženskaja], V. A. Solonnikov and N. N. Ural\(^{\prime }\)tseva [N. N. Ural’ceva], “Linear and Quasilinear Equations of Parabolic Type,” translated from the Russian by S. Smith, Transl. Math. Monogr., Vol. 23, American Mathematical Society, Providence, RI, 1967.
Z. Lian and K. Lu, “Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space,” Mem. Amer. Math. Soc. 206 (2010), no. 967.
R. H. Martin, Jr., and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc. 321 (1990), no. 1, 1–44.
R. H. Martin, Jr., and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence, J. Reine Angew. Math. 413 (1991), 1–35.
J. Mierczyński, S. Novo and R. Obaya, Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equations., Discrete Contin. Dyn. Syst. 38 (2018), no. 12, 6163–6193.
J. Mierczyński, S. Novo and R. Obaya, Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations, Commun. Pure Appl. Anal. 19 (2020), no. 4, 2235–2255
J. Mierczyński and W. Shen, “Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,” Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2008.
J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. III. Parabolic equations and delay systems, J. Dynam. Differential Equations 28 (2016), no. 3–4, 1039–1079.
S. Novo, C. Núñez, R. Obaya and A. M. Sanz, Skew-product semiflows for non-autonomous partial functional differential equations with delay, Discrete Contin. Dyn. Syst. 34 (2014), no. 10, 4291–4321.
R. Obaya and A. M. Sanz, Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 8, 3947–3970.
R. J. Sacker, A new metric yielding a richer class of unbounded functions having compact hulls in the shift flow, J. Dynam. Differential Equations 33 (2021), no. 2, 833–848.
R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations 113 (1994), no. 1, 17–67.
W. Shen and Y. Yi, “Almost Automorphic and Almost Periodic Dynamics in Skew-product Semiflows,” Mem. Amer. Math. Soc. 136 (1998), no. 647.
C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395–418.
C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable, J. Math. Anal. Appl. 56 (1976), no. 2, 397–409.
C. C. Travis and G. F. Webb, Existence, stability, and compactness in the\(\alpha \)-norm for partial functional differential equations, Trans. Amer. Math. Soc. 240 (1978), 129–143.
V. S. Varadarajan, On the convergence of sample probability distributions, Sankhyā 19 (1958), no. 1–2, 23–26.
J. Wu, “Theory and Applications of Partial Functional-Differential Equations,” Appl. Math. Sci., 119, Springer, New York, 1996.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kryspin, M., Mierczyński, J. Parabolic differential equations with bounded delay. J. Evol. Equ. 23, 2 (2023). https://doi.org/10.1007/s00028-022-00848-w
Accepted:
Published:
DOI: https://doi.org/10.1007/s00028-022-00848-w
Keywords
- Linear parabolic partial differential equation with delay
- Existence and uniqueness of solutions
- Continuous dependence of solutions on parameters