1 Introduction and preliminaries

In this paper, we investigate the existence and regularity properties of the solutions to the evolution problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A(t)u(t)= f(t,u(t)), \qquad a.e. \quad t\in [0,T]\ \\ u(0)= g(u) \end{array}\right. } \end{aligned}$$
(1.1)

where A(t) is a non-autonomous operator having \(L^2\)-maximal regularity, f is nonlinear and demicontinuous in the second variable and g is a function representing a non-local condition.

Maximal regularity results have been the subject of several recent studies. On one side, this is due to the fact that the maximal regularity property of the solutions of differential equations gains a central role in the theory of parabolic problems, by permitting weaker requests on the regularity of the coefficients of differential operators. On the other hand, from an abstract point of view, maximal regularity characterizes a wide class of evolution equations and facilitates the application of linearization techniques.

This property plays an essential part indeed, together with a finite dimensional reduction technique and the Leray–Schauder continuation principle, permit us to prove the existence and regularity properties of the solutions to a wide class of non-autonomous semilinear evolution equations, including demicontinuous nonlinearities.

Turning our attention to the trace space, we should note that while in the autonomous case, the theory is well-developed and it is well known in the literature in which spaces must be chosen in order to reach the desired regularity of the solutions, the non-autonomous setting had been the subject of several recent research papers.

The evolution problem is then completed by an initial condition of non-local type; it is remarkable that non-local conditions can be more useful than standard initial conditions \(u(0)=u_0\) to model some physical phenomena (see [30, 40] and references therein).

To be more precise, let \((V,\langle \cdot , \cdot \rangle _V)\) and \((H,\langle \cdot , \cdot \rangle _H)\) be two separable Hilbert spaces such that V is continuously and densely embedded into H, i.e., V is a dense subspace of H such that

$$\begin{aligned} \Vert v\Vert _H \le c \Vert v \Vert _V \end{aligned}$$

for some constant \(c>0\) and any \(v\in V.\) In many practical examples, an operator \({\mathcal {A}} (t)\) is associated to a bounded sesquilinear form \(a(t,\cdot ,\cdot )\) with domain V.

More precisely, assume that \(a:[0,T]\times V \times V \rightarrow {\mathbb {C}}\) satisfies

  1. (H1)

    \(a(\cdot ,u,v)\) is strongly measurable for any \(u,v\in V,\)

  2. (H2)

    There exists \(M>0\) such that \(|a(t,u,v)|\le M \Vert u \Vert _V \Vert v\Vert _V\) for any \(t\in [0,T]\) and \(u,v\in V,\)

  3. (H3)

    There exists \(\alpha >0\) such that \({{\mathfrak {R}}}{{\mathfrak {e}}}( a(t,u,u) ) \ge \alpha \Vert u\Vert ^2_V\) for any \(t\in [0,T]\) and \(u\in V;\)

Then, for any \(t\in [0,T],\) \({\mathcal {A}}(t)\in {\mathcal {L}}(V,V')\) is well-defined by \(\langle {\mathcal {A}}(t) u, v\rangle _{V' \times V} := a(t,u,v) \) and \({\mathcal {D}}({\mathcal {A}}(t))=V,\) where \(\langle \cdot , \cdot \rangle _{V' \times V}\) is the standard duality pairing.

Turning our attention to evolution problems governed by forms, we mention the following result proved by J.L. Lions in 1961 (see [31]):

Theorem 1.1

For any fixed \(x\in H\) and \(f\in L^2([0,T],V'),\) the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+{\mathcal {A}}(t)u(t)= f(t), \quad a.e. \quad t\in [0,T]\ \\ u(0)=x, \end{array}\right. } \end{aligned}$$
(1.2)

has a unique solution \(u\in L^2([0,T],V)\cap H^1([0,T],V').\)

We point out that the outstanding result by Lions only requires the measurability of \(a(\cdot ,u,v)\). On the other side, the above result is not fully satisfactory when applied to boundary value problems. Indeed, it is the part of \({\mathcal {A}}(t)\) which lies in H that realizes the boundary conditions. Therefore, let \(A(t),\ t\in [0,T]\) be defined by \(A(t) u:={\mathcal {A}} (t) u,\) on the non-empty set \({\mathcal {D}}(A(t)):=\{u\in V: {\mathcal {A}} (t) u \in H\}\) and focus on

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+{A}(t)u(t)= f(t), \quad a.e. \quad t\in [0,T]\ \\ u(0)=x. \end{array}\right. } \end{aligned}$$
(1.3)

Definition 1.2

For fixed x in a suitable trace space, (1.3) is said to have \(L^2\)-maximal regularity in H if for any \(f\in L^2([0,T],H)\) there exists a unique solution \(u\in H^1 ([0,T],H) \cap L^2([0,T],V)\) and such that \(u(t)\in {\mathcal {D}}(A(t)).\)

Several authors dealt with the problem of establishing which conditions on the trace space and which regularity assumptions on a(t, u, v) are sufficient to achieve maximal regularity in H. Among others, we cite the papers [2, 3, 44].

In 2015, Haak and Ouhabaz [27] gave a complete treatment of the evolution problem with non-zero initial values. We will mention their general result which we restate to our scope.

To this end, note that condition (H3) readily implies that A(t) is accretive for any \(t\in [0,T],\) i.e.,

$$\begin{aligned} {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A(t) u, u \rangle _H \ge 0 \text { for any } u\in {\mathcal {D}}(A(t)). \end{aligned}$$

Secondly, assume that a(t, u, v) also satisfies

  1. (H4)

    \(|a(t,u,v)-a(s,u,v)|\le \omega (|t-s|) \Vert u \Vert _V \Vert v \Vert _V,\) for some nondecreasing \(\omega :[0,T]\rightarrow [0,+\infty )\) which satisfies

    $$\begin{aligned} \int _0^T \frac{\omega (t)}{t^{3/2}} dt< \infty \text { and the Dini condition } \int _0^T \left( \frac{\omega (t)}{t} \right) ^2 dt < \infty . \end{aligned}$$

Theorem 1.3

[27, Corollary 3] Suppose that (H1)–(H4) and \({\mathcal {D}}(A(0)^{1/2})=V\) are satisfied, then for any \(x\in V\) and any \(f\in L^2([0,T],H),\) (1.3) has \(L^2\)-maximal regularity in H. Moreover, there exists a constant \(C_0>0\) such that

$$\begin{aligned} \Vert u \Vert _{H^1 ([0,T],H)} + \Vert A(\cdot ) u(\cdot ) \Vert _{L^2([0,T],H)} \le C_0 \left( \Vert f \Vert _{L^2([0,T],H)} + \Vert x \Vert _V \right) . \end{aligned}$$
(1.4)

To our scope, we need to introduce a stronger condition on the domain, that is we assume that

  1. (S)

    for any \(t\in [0,T],\) the square root property \({\mathcal {D}}(A(t)^{1/2}) = V\) holds.

Three remarkable facts are detailed below. The first of them shows an immediate consequence of the above estimate, while the others two illustrate settings in which conditions (H3) and (S) can be lowered.

Remark 1.4

We stress that condition (H3) guarantees that the solution u lies in \(L^2([0,T],V).\) More precisely, there exists \(C_1>0\) such that

$$\begin{aligned}&\Vert u \Vert _{H^1 ([0,T],H)} + \Vert u \Vert _{L^2([0,T],V)} \nonumber \\&\quad + \Vert A(\cdot ) u(\cdot ) \Vert _{L^2([0,T],H)} \le C_1 \left( \Vert f \Vert _{L^2([0,T],H)} + \Vert x \Vert _V \right) . \end{aligned}$$
(1.5)

Indeed, observe that

$$\begin{aligned} \alpha \Vert u(t) \Vert ^2_V&\le {{\mathfrak {R}}}{{\mathfrak {e}}}\left( a(t,u(t),u(t)) \right) = {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A(t) u(t), u(t) \rangle _H \\&\le \frac{1}{2} \left( \Vert u(t) \Vert ^2_H + \Vert A(t)u(t)\Vert ^2_H \right) , \end{aligned}$$

which implies, by means of (1.4) that

$$\begin{aligned} \Vert u \Vert ^2_{L^2([0,T],V)}\le & {} \frac{1}{2\alpha } \left( \Vert u \Vert ^2_{L^2([0,T],H)} + \Vert A(t)u(t)\Vert ^2_{L^2([0,T],H)} \right) \\\le & {} \frac{C_0^2}{2\alpha } \left( \Vert f \Vert _{L^2([0,T],H)} + \Vert x \Vert _V \right) ^2. \end{aligned}$$

Hence, (1.5) follows by setting \(C_1:= C_0 \left( 1 + \left( \frac{1}{2\alpha } \right) ^{\frac{1}{2}}\right) .\)

Remark 1.5

Let \(\mu >0\) and set \(v(t):=e^{-\mu t} u(t).\) Then, if u satisfies problem (1.3), then v is the unique solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} v'(t)+{A}(t)v(t) + \mu v(t) =g(t), \quad a.e. \quad t\in [0,T] \\ v(0)=u(0), \end{array}\right. } \end{aligned}$$

where \(g(t):=e^{-\mu t} f(t).\) This shows that condition (H3) can be lowered by assuming that

(H3\(^*\)):

There exists \(\alpha ,\delta >0\) such that

$$\begin{aligned} \delta \Vert u \Vert ^2_H + {{\mathfrak {R}}}{{\mathfrak {e}}}( a(t,u,u) ) \ge \alpha \Vert u\Vert ^2_V \end{aligned}$$

holds for any \(t\in [0,T]\) and \(u\in V.\)

Remark 1.6

Assume that

(H4\(^*\)):

\(|a(t,u,v)-a(s,u,v)|\le \omega (|t-s|) \Vert u \Vert _V \Vert v \Vert _{V_\gamma }\)

holds for some bounded \(\omega \) and where \(V_\gamma \) is the complex interpolation space \([H,V]_\gamma \) for a fixed \(\gamma \in (0,1).\) Then, it follows from [6, Proposition 2.5] that property (S) can be relaxed by assuming

(S\(^*\)):

\({\mathcal {D}}(A(t_0)^{1/2}) = V\) for some \(t_0\in [0,T].\)

Based on Theorem 1.3 and by closely following the lines provided by [2], in Sect. 2 we construct a strongly continuous evolution family and introduce some basic properties which will be crucial in the sequel.

In several recent papers (see e.g., [11, 33, 50] and references therein), a final dimensional reduction strategy had been proved to be a key point when associated to the study of autonomous evolution problems. In particular, a new method has been introduced in [13] which combines Yosida approximation with the approximation solvability method. The aforementioned method is a generalization of the well-known strong approximation technique ([18, 43]) and has been used in [32] for studying periodic oscillations, while in [12] it has been applied to nonlocal differential equation problems in Banach spaces. In this article, we show that the technique also fits well in the non-autonomous case and, as a novelty element, we combine the approximation solvability method with the approximation properties of evolution systems governed by forms. Section 3 is devoted to the scope.

Indeed, we prove that Theorem 1.3 also applies to evolution equations governed by operator of type \({\mathbb {P}} A(t),\) where \({\mathbb {P}}\) is the projection onto a finite dimensional space (Lemma 3.3). An approximation result is also introduced to show the uniform convergence of the evolution systems generated by \({\mathbb {P}} A(t)\) (Lemma 3.4).

The above-mentioned approach permits us to lower the hypothesis on the nonlinear term, as explained below.

Let X and Y be Banach spaces and let \(f:[0,T]\times X \rightarrow Y\) be a map. An important concept in Functional Analysis is the one of superposition operator \(N_f :L^p ([0,T],X) \rightarrow L^q([0,T],Y)\), defined by \(N_f (u) (t) := f(t, u(t)).\) We recall the following classical theorem:

Theorem 1.7

([34]) If X and Y are separable and f is measurable in \([0,T]\times X,\) then \(N_f:L^p ([0,T],X) \rightarrow L^q([0,T],Y)\) is well-defined if and only if there exists a constant \(a>0\) and a function \(b\in L^q ([0,T],{\mathbb {R}}_+)\) such that

$$\begin{aligned} \Vert f(t,x) \Vert _Y \le a \Vert x \Vert _X^{p/q} + b(t). \end{aligned}$$
(1.6)

Moreover, \(N_f\) maps bounded subsets into bounded subsets.

In classical examples on \(L^p\) spaces, nonlinear superposition operators are often constructed on Caratheodory maps \(f:[0,T]\times X \rightarrow Y\), i.e., by assuming that f is continuous w.r.t. the second variable. The continuity of the superposition operator \(N_f\) is then guaranteed as a consequence of this fact.

We stress that the continuity of superposition operator is almost a necessary property in the applications to differential systems. Nevertheless, the use of weak topologies can often be more convenient in several practical problems. Unfortunately, [37, Example 2.3] shows that weak continuity of superposition operator \(N_f\) acting on \(L^p\) spaces does not derive from the same assumption on the map \(f(t,\cdot ).\)

Therefore, a stronger form of continuity on \(f(t,\cdot )\) appears to be necessary in order for the superposition operator to continuously act on spaces endowed with the weak topology (see [37, Theorem 2.6]).

In Sect. 4, we show that the operator \(N_f\) is demicontinuous or, in other words, that for any sequence \(x_n \rightarrow x\) it holds \(N_f (x_n) \rightharpoonup N_f(x),\) whenever \(f(t,\cdot )\) satisfies the same property (Lemma 4.2).

Demicontinuity fits well with the finite dimensional reduction approach and permit us, under the further assumption that the map f satisfies a transversality condition, to apply the Leray-Schauder continuation principle (see also [10]):

Theorem 1.8

([29]) Let D be a closed and convex set in a Banach space X. Let \(S:[0,1]\times D \rightarrow X\) be a completely continuous operator (i.e., S maps bounded subsets into compact subsets and it is continuous) and assume that

  1. (i)

    \(S(0,x)\in {\text {int}} (D)\) for any \(x\in D;\)

  2. (ii)

    The set

    $$\begin{aligned} \{ x \in D : x = S(\lambda , x) \text { for some } \lambda \in [0,1] \} \end{aligned}$$

    is bounded and does not meet the boundary \(\partial D\) of D.

Then there exists \(x\in D\) such that \(x=S(1,x).\)

Finally, a solution to problem (1.1) is proved to exist as a limit. For this last step, the following result is crucial.

Theorem 1.9

(Aubin–Lions Lemma, see e.g., [14]) Let \(X_0,X\) and \(X_1\) be three Banach spaces. Suppose that \(X_0\) is compactly embedded in X and X is continuously embedded in \(X_1\). Suppose also that \(X_0\) and \(X_1\) are reflexive. Then for \(0<T<+\infty \) and \(1<r,s<\infty ,\) we have that \(L^r([0,T],X_0)\cap W^{1,s} ([0,T],X_1)\) is compactly embedded in \(L^r([0,T],X).\)

The results detailed above are contained in Sect. 5.

The paper is then concluded by Sect. 6, where application examples are discussed. In particular, we exploit the fact that the demicontinuity assumption is naturally satisfied by any maximally monotone operator with sublinear growth.

We stress that the novelty elements that we introduce in this manuscript lie in the approximation results (Lemmas 3.3 and 3.4), in Lemma 4.2 where the demicontinuity of the superposition operator is shown and, leastly, in Theorem 5.1 we show the existence of solutions in \(H^1([0,T],H) \cap L^2([0,T],V)\) to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A(t)u(t)=f(t,u(t)) \qquad t\in [0,T]\ \\ u(0)=g(u). \end{array}\right. } \end{aligned}$$
(1.7)

It is worth mentioning that another novelty element of our approach lies in the regularity properties of the solutions to the nonlocal semilinear problem. Indeed, it is important to note that the majority of the papers on the topic mainly deal with the existence of mild solutions, that is continuous solutions of an associated fixed-point problem. Here we prove that the aforementioned approximation solvability method can be a key-strategy also for proving the existence of strong solutions.

2 Evolution families for operators governed by forms

In this section, we study the properties of the evolution family generated by a non-autonomous form which satisfies properties (H1)–(H4) and (S).

Here and through the rest of the paper, \((V,\langle \cdot , \cdot \rangle _V)\) and \((H,\langle \cdot , \cdot \rangle _H)\) will be two separable Hilbert spaces, with V densely embedded into H and \(\Delta \) will denote the set \(\{(t,s): 0\le s < t \le T\}.\)

As usual, \(L^2([0,T],H)\) is the Lebesgue space of all the square integrable H-valued functions, while \(H^1([0,T],H)=W^{1,2}([0,T],H)\) is the corresponding Sobolev space.

We turn our attention to the construction of a suitable evolution family by introducing the next lemma, a direct consequence of Theorem 1.3, which proof follows the line of Propositions 2.3 and 2.4 and Corollaries 3.4 and 3.5 from [2]. Our hypotheses are slightly different and a proof is provided for sake of completeness. Also we use \(C\in (0,+\infty )\) to denote any limiting constant, agreeing since now to redefine it whenever it is needed.

Lemma 2.1

Assume that (H1)–(H4) and (S) are satisfied, then there exists a contractive and strongly continuous evolution family \(\{E(t,s)\}_{(t,s)\in \Delta } \subset {\mathcal {L}} (H)\) such that

  1. (i)

    \(u(t):=E(t,s)x\) is the unique solution in \(H^1 ([s,T],H) \cap L^2{([s,T],V)}\) of the homogeneous problem

    $$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+{A}(t)u(t)= 0, \quad a.e. \quad t\in [s,T]\ \\ u(s)=x\in V, \end{array}\right. } \end{aligned}$$
    (2.1)

    moreover it holds

    $$\begin{aligned} \Vert u \Vert _{H^1 ([s,T],H)} + \Vert u \Vert _{L^2([s,T],V)} + \Vert A(\cdot ) u(\cdot ) \Vert _{L^2([s,T],H)} \le C \Vert x\Vert _V \end{aligned}$$
    (2.2)

    where \(C>0\) is a constant;

  2. (ii)

    \(u(t):=E(t,0)x\) is the unique solution in \(H^1_\text {loc} ((0,T],H) \cap L^2_\text {loc}((0,T],V)\cap C([0,T],H)\) of the homogeneous problem

    $$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+{A}(t)u(t)= 0, \quad a.e. \quad t\in [0,T]\ \\ u(0)=x\in H, \end{array}\right. } \end{aligned}$$

    and it holds

    $$\begin{aligned} \Vert v\Vert _{H^{1}([0,T],H)}+\Vert v\Vert _{L^2([0,T],V)} +\Vert A(\cdot )v(\cdot )\Vert _{L^2([0,T],H)} \le C\sqrt{T}\Vert x\Vert _{H}, \end{aligned}$$
    (2.3)

    where \(v(t):=tE(t,0)x.\)

  3. (iii)

    For any \(x\in H\) and \(f\in L^2([0,T],H),\) the function

    $$\begin{aligned} E(t,0)x+\int _0^t E(t,s) f(s) ds \end{aligned}$$
    (2.4)

    is the unique solution in \(H^1_\text {loc} ((0,T],H) \cap L^2_\text {loc}((0,T],V)\cap C([0,T],H)\) of (1.3).

  4. (iv)

    For any \(x\in V\) and \(f\in L^2([0,T],H),\) the unique solution u of (1.3) given by Theorem 1.3 has a continuous representation as

    $$\begin{aligned} u(t) = E(t,0)x+\int _0^t E(t,s) f(s) ds. \end{aligned}$$

Proof

From Theorem 1.3 and by [2, Proposition 2.3], a strongly continuous evolution family \(\{E_0(t,s)\}_{(t,s)\in \Delta }\subset {\mathcal {L}}(V)\) exists such that \(u(t):=E_0(t,s)x\) is the unique solution in \(H^1 ([s,T],H) \cap L^2([s,T],V)\) to the Cauchy problem (2.1).

Since A(t) is accretive, it follows that

$$\begin{aligned} \Vert u(t) \Vert _H^2 - \Vert u(s)\Vert _H^2&= \int _s^t \frac{d}{dt} \Vert u(\tau ) \Vert _H^2 d\tau = 2 \int _s^t {{\mathfrak {R}}}{{\mathfrak {e}}}\langle u'(\tau ) , u(\tau ) \rangle _H d\tau \\&= 2 \int _s^t - {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A(\tau ) u(\tau ), u(\tau ) \rangle _H d\tau \le 0, \end{aligned}$$

which readily implies that, for any fixed \((t,s)\in \Delta \),

$$\begin{aligned} \Vert E_0(t,s) x \Vert _H = \Vert u(t)\Vert _H \le \Vert u(s)\Vert _H = \Vert x \Vert _H. \end{aligned}$$

This last inequality, together with the density of V in H, permits to uniquely extend \(E_0(t,s): V \rightarrow H\) to a linear operator \(E(t,s):H\rightarrow H\) by means of the Bounded Linear Transformation Theorem (see e.g., [36]). The strong continuity of the family \(\{ E(t,s) \}_{(t,s)\in \Delta }\) also follows from a density argument.

  1. (i)

    The claims follow from above construction and (1.5) of Remark 1.4.

  2. (ii)

    Arguing as in [2, Corollary 3.4], we start by fixing \(x\in V.\) Set \(v(t):= t E(t,0)x\) then, since \(E(\cdot , 0)x\in L^2([0,T],H)\) and by Theorem 1.3, v is the unique solution of the non-homogeneous Cauchy problem

    $$\begin{aligned} {\left\{ \begin{array}{ll} v'(t)+{A}(t)v(t)=E (t,0)x, \qquad a.e. \quad t\in [0,T] \\ v(0)=0, \end{array}\right. } \end{aligned}$$

    moreover by (1.5),

    $$\begin{aligned}&\Vert v\Vert _{H^{1}([0,T],H)}+\Vert v\Vert _{L^2([0,T],V)} +\Vert A(\cdot )v(\cdot )\Vert _{L^2([0,T],H)}\nonumber \\&\quad \le C\left( \Vert E(\cdot ,0)x\Vert _{L^2([0,T],H)}\right) \nonumber \\&\quad \le C\sqrt{T}\Vert x\Vert _{H}. \end{aligned}$$
    (2.5)

    Observe that

    $$\begin{aligned} \varepsilon ^2 \left\| E (\cdot ,0)x \right\| ^2_{L^2([\varepsilon ,T],H)}= & {} \varepsilon ^2 \int _\varepsilon ^T \left\| E (t,0)x \right\| _H^2 dt \le \int _\varepsilon ^T \left\| t E (t,0)x \right\| _H^2 dt \\\le & {} \Vert v \Vert ^2_{L^2([0,T],H) }. \end{aligned}$$

    Similarly \( \left\| E (\cdot ,0)x \right\| _{L^2([\varepsilon ,T],V)} \Vert \le \varepsilon ^{-1} \Vert v \Vert _{L^2([0,T],V) }.\) A last estimate is reached by considering the following

    $$\begin{aligned}&\varepsilon ^2 \left\| \frac{dE (\cdot ,0)x}{dt}\right\| ^2_{L^2([\varepsilon ,T],H)}\nonumber \\&\quad \le \int _\varepsilon ^T \left\| t \frac{dE (t,0)x}{dt} \right\| ^2_H dt\\&\quad \le \int _\varepsilon ^T \left( \left\| t \frac{dE (t,0)x}{dt} + E (t,0)x \right\| _H + \Vert E (t,0)x \Vert _H \right) ^2 dt\\&\quad \le 2 \Vert v'\Vert ^2_{L^2([0,T],H)} + 2\Vert E(\cdot ,0)x\Vert ^2_{L^2([\varepsilon ,T],H)}. \end{aligned}$$

    Putting the last three inequalities together with (2.5) and by redefining the constant \(C>0,\) we obtain that

    $$\begin{aligned}&\Vert E(\cdot ,0)x\Vert _{L^2([\varepsilon ,T],H)}+\Vert E(\cdot ,0)x\Vert _{L^2([\varepsilon ,T],V)}\\&\quad +\left\| \frac{dE (\cdot ,0)x}{dt}\right\| _{L^2([\varepsilon ,T],H)} \le \frac{C\sqrt{T}}{\varepsilon }\Vert x \Vert _H \end{aligned}$$

    holds for any fixed \(\varepsilon \in (0,T].\) By exploiting again the density of V in H,  it follows that the same inequalities also holds for any fixed \(x\in H.\) We have then proved that E(t, 0)x lies in \(H^1_\text {loc} ((0,T],H) \cap L^2_\text {loc}((0,T],V)\cap C([0,T],H)\) as well as the estimate (2.3).

  3. (iii)

    It essentially follows as in [2, Proposition 2.4] for \(p=2.\)

  4. (iv)

    By maximal regularity, u has a continuous representation in C([0, T], H) (see [46]) and u solves (1.3) in \(H^1_\text {loc} ((0,T],H) \cap L^2_\text {loc}((0,T],V)\cap C([0,T],H).\) Then result derives from the uniqueness of the solution.\(\square \)

3 Finite dimensional reduction

Let \(\{ \varphi _n\}_{n\in {\mathbb {N}}}\) be a Schauder basis for V and hence for H since V densely embeds onto H. We assume \(\{ \varphi _n\}_{n\in {\mathbb {N}}}\) to be orthogonal w.r.t. the inner product of H,  \(\langle \cdot , \cdot \rangle _H .\) For a fixed \(m\in {\mathbb {N}},\) let \({\mathbb {P}}:H \rightarrow {\text {span}}_{\mathbb {C}}\{\varphi _j : 1\le j\le m\}\) be the projection \({\mathbb {P}} (\sum _{j=1}^\infty v_j \varphi _j) := \sum _{j=1}^m v_j \varphi _j.\) To avoid confusion, by \(H_m\) we will denote the space \({\mathbb {P}} H\) endowed with the norm \(\Vert \cdot \Vert _H,\) while \(V_m\) indicates that the equivalent norm \(\Vert \cdot \Vert _V\) is considered on the same finite dimensional vector space.

Note that \({\mathbb {P}}\) is self-adjoint with respect to the H-inner product (that is w.r.t. \(\langle \cdot , \cdot \rangle _H\)), indeed whenever \(u,v \in H,\) we have

$$\begin{aligned} \left\langle u, {\mathbb {P}} v \right\rangle _H = \lim _{k\rightarrow \infty } \left\langle \sum _{i=1}^k u_i \varphi _i, \sum _{j=1}^m v_j \varphi _j \right\rangle _H = \left\langle \sum _{i=1}^m u_i \varphi _i, \sum _{j=1}^m v_j \varphi _j \right\rangle _H = \left\langle {\mathbb {P}} u, v \right\rangle _H. \end{aligned}$$

Of course, since \(\{ \varphi _n\}_{n\in {\mathbb {N}}}\) is a Schauder basis for both V and H,  it holds that \(\Vert v - {\mathbb {P}} v\Vert _V \rightarrow 0\) and \(\Vert x-{\mathbb {P}} x \Vert _H \rightarrow 0\) as \(m\rightarrow \infty \) for any fixed \(v\in V\) and \(x\in H.\) Let \(a:[0,T] \times V\times V \rightarrow {\mathbb {C}}\) be a sesquilinear form satisfying (H1)–(H4). For fixed \(m\in {\mathbb {N}}\), we denote with \(a_m : [0,T] \times V\times V \rightarrow {\mathbb {C}}\) the sesquilinear form given by

$$\begin{aligned} a_m (t,u.v) := a(t,{\mathbb {P}} u, {\mathbb {P}} v) + \alpha \langle (I-{\mathbb {P}})u , (I-{\mathbb {P}})v \rangle _V. \end{aligned}$$

Remark 3.1

Note that \(a_m\) also satisfies properties (H1)–(H4). Indeed, (H1) and (H4) are trivially satisfied while (H2) is easily derived from

$$\begin{aligned} | a_m (t,u,v) |\le & {} M \Vert {\mathbb {P}} u \Vert _V \Vert {\mathbb {P}} v\Vert _V + \alpha \Vert (I - {\mathbb {P}}) u \Vert _V \Vert (I-{\mathbb {P}}) v\Vert _V \\\le & {} k_m (M+\alpha )\Vert u\Vert _V \Vert v\Vert _V, \end{aligned}$$

since both \(\Vert {\mathbb {P}} \Vert _{{\mathcal {L}}(V)}\) and \(\Vert I-{\mathbb {P}} \Vert _{{\mathcal {L}}(V)}\) are bounded by some \(k_m >0\) for any fixed \(m\in {\mathbb {N}}.\)

To prove (H3) observe that

$$\begin{aligned} {{\mathfrak {R}}}{{\mathfrak {e}}}( a_m(t,u,u) )= & {} {{\mathfrak {R}}}{{\mathfrak {e}}}( a (t, {\mathbb {P}} u, {\mathbb {P}} u) )+ \alpha \Vert (I - {\mathbb {P}}) u \Vert _V^2 \\\ge & {} \alpha \Vert {\mathbb {P}} u \Vert _V^2 + \alpha \Vert (I - {\mathbb {P}}) u \Vert _V^2\\= & {} 2 \alpha \left( \frac{1}{2} \Vert {\mathbb {P}} u \Vert _V^2 + \frac{1}{2} \Vert (I - {\mathbb {P}} )u \Vert _V^2 \right) \\\ge & {} 2 \alpha \left\| \frac{1}{2} {\mathbb {P}} u + \frac{1}{2} u -\frac{1}{2} {\mathbb {P}} u\right\| _V^2\\= & {} \frac{\alpha }{2} \Vert u \Vert _V^2 \end{aligned}$$

since \(\Vert \cdot \Vert _V^2\) is convex.

Let \(A_m (t)\) be the operator defined by

$$\begin{aligned} a_m(t,u,v) := \langle A_m (t) u, v \rangle _H \text { for all }v\in V \end{aligned}$$

and with domain \({\mathcal {D}}(A_m (t)):=\{ u \in V: A_m (t) u \in H\}.\)

Remark 3.2

Let \(B:V \rightarrow H\) be the operator associated to the sesquilinear form \(\langle \cdot , \cdot \rangle _V.\) Then for any fixed \(t\in [0,T]\) and for any \(u,v\in V\), one has

$$\begin{aligned} \langle A_m (t) u, v\rangle _H= & {} a_m(t,u,v) = a(t,{\mathbb {P}} u, {\mathbb {P}} v) + \alpha \langle (I-{\mathbb {P}}) u, (I-{\mathbb {P}}) v \rangle _V \\= & {} \langle A (t){\mathbb {P}} u, {\mathbb {P}} v \rangle _H + \alpha \langle B (I-{\mathbb {P}}) u, (I-{\mathbb {P}}) v \rangle _H \\= & {} \langle ({\mathbb {P}} A(t) {\mathbb {P}} +\alpha \langle (I-{\mathbb {P}}) B (I-{\mathbb {P}}))u,v\rangle _H. \end{aligned}$$

Therefore, the identity

$$\begin{aligned} A_m (t) = {\mathbb {P}} A(t) {\mathbb {P}} + \alpha (I-{\mathbb {P}})B(I-{\mathbb {P}}) \end{aligned}$$
(3.1)

follows by density. It is then straightforward to see that for any \(u\in V_m,\)

$$\begin{aligned} A_m(t) u = {\mathbb {P}} A(t) u \end{aligned}$$

holds.

Lemma 3.3

For any fixed \(m\in {\mathbb {N}},\) \(A_m(t)\) defined above generates a contractive evolution system \(\{E_m (t,s) \} \subset {\mathcal {L}}(V)\) such that for any \(x\in V,\) \(u(t):=E_m(t,s)x\) is the unique solution in \(H^1 (s,T,H) \cap L^2(s,T,V)\) of the homogeneous problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+{A_m}(t)u(t)= 0, \qquad a.e. \quad t\in [s,T]\ \\ u(s)=x \in V, \end{array}\right. } \end{aligned}$$
(3.2)

which satisfies inequality (2.2). Moreover, if \(x\in V_m\) then \(u(t)\in V_m\) for any \(t\in [0,T].\)

Proof

The existence of the evolution system \(\{ E_m (t,s) \}\) will directly follow from Lemma 2.1 once we prove that \(A_m(t)\) satisfies property (S).

Fix \(t\in [0,T].\) It follows from [5, Corollary 7.1.4] that the operator \(A_m(t) = {\mathbb {P}} A(t) {\mathbb {P}} + \alpha (I-{\mathbb {P}}) B (I-{\mathbb {P}})\) generates a cosine function on H,  since \({\mathbb {P}} A(t) {\mathbb {P}}\) is bounded and \(\alpha (I-{\mathbb {P}})B(I-{\mathbb {P}})\) is symmetric.

By [28, Corollary 5.18], the numerical range \(W(A_m(t)):=\{ \langle A_m(t) u, u \rangle _H | u\in V, \Vert u \Vert _H=1\}\) is then contained in a parabola. Lastly, by [35, Theorems A and C] it follows that the square root condition \({\mathcal {D}}(A_m(t)^{1/2} )=V\) holds (see also [4]). This proves the first part.

Let \(x\in V_m\) and let u(t) be the solution of (3.2). Note that the function \(t\rightarrow u(t)-{\mathbb {P}} u(t)\) has a.e. derivative \( u'(t) -{\mathbb {P}} u'(t)\) since \({\mathbb {P}}\) is linear; moreover, it holds

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert u(t) - {\mathbb {P}} u(t) \Vert _H^2= & {} {{\mathfrak {R}}}{{\mathfrak {e}}}\langle u' (t) - {\mathbb {P}} u' (t), u(t)- {\mathbb {P}} u(t) \rangle _H \nonumber \\= & {} - {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A_m (t) u(t), u(t)-{\mathbb {P}} u(t) \rangle _H \nonumber \\&+ {{\mathfrak {R}}}{{\mathfrak {e}}}\langle {\mathbb {P}} A_m (t) u(t), u(t)-{\mathbb {P}} u(t) \rangle _H\nonumber \\= & {} - {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A_m (t) u(t), u(t)-{\mathbb {P}} u(t) \rangle _H \nonumber \\&+ {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A_m (t) u(t), {\mathbb {P}} ( u(t)-{\mathbb {P}} u(t) ) \rangle _H \nonumber \\= & {} -{{\mathfrak {R}}}{{\mathfrak {e}}}\langle A_m (t) u(t), u(t)-{\mathbb {P}} u(t) \rangle _H \nonumber \\= & {} - a_m (t, u (t), u(t) - {\mathbb {P}} u (t) )\nonumber \\= & {} -a (t, {\mathbb {P}} u(t), {\mathbb {P}} ( I-{\mathbb {P}}) u(t)) \nonumber \\&- \alpha \Vert (I-{\mathbb {P}}) u(t) \Vert _V^2, \end{aligned}$$
(3.3)

where we have used the fact that \({\mathbb {P}}\) is self-adjoint w.r.t \(\langle \cdot , \cdot \rangle _H\) and \(I-{\mathbb {P}}\) is idempotent. Note that \({\mathbb {P}} (I-{\mathbb {P}}) = 0,\) which implies that \(a (t, {\mathbb {P}} u(t), {\mathbb {P}} ( I-{\mathbb {P}}) u(t))=0\) and (3.3) brings to

$$\begin{aligned} \frac{d}{dt} \Vert u(t) - {\mathbb {P}} u(t) \Vert _H^2 \le 0 - \alpha \Vert (I-{\mathbb {P}}) u(t) \Vert _V^2 \le 0. \end{aligned}$$

The direct consequence of this last is that for a.e. \(t\in [0,T],\) \( \Vert u(t) - {\mathbb {P}} u(t)\Vert _H \le \Vert x - {\mathbb {P}} x\Vert _H=0\) since \(x\in V_m;\) thus, \(u(t)={\mathbb {P}} u(t)\) and \(u(t)\in V_m\) follows. \(\square \)

Lemma 3.4

Let \(\{E(t,s)\}_{(t,s)\in \Delta }\) and \(\{E_{m}(t,s)\}_{(t,s)\in \Delta }\) be the evolution systems generated by A(t) and \(A_{m}(t)\), respectively. Then for any fixed \(x\in H,\) \(\{ E_{m}(t,s){\mathbb {P}} x\}_{n\in {\mathbb {N}}}\) converges in H to E(t, s)x uniformly on \(t>s\) in [0, T].

Proof

We start by fixing \(x\in V\) and, for fixed \((t,s)\in \Delta ,\) set \(u(t):=E(t,s)x,\) \(u_m(t):=E_m(t,s){\mathbb {P}} x\) and \(z_m(t):=u_m(t)-u(t).\) Note that by definition,

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \Vert z_m (t) \Vert _H ^2= & {} -{{\mathfrak {R}}}{{\mathfrak {e}}}\langle \ {\mathbb {P}} A (t) u_m (t), u_m (t) - u(t) \rangle _H \nonumber \\&+{{\mathfrak {R}}}{{\mathfrak {e}}}\langle A (t) u(t), u_m (t)-u(t) \rangle _H \nonumber \\= & {} -{{\mathfrak {R}}}{{\mathfrak {e}}}\langle A (t) u_m (t), {\mathbb {P}} u_m (t) - {\mathbb {P}} u(t) \rangle _H \nonumber \\&+{{\mathfrak {R}}}{{\mathfrak {e}}}\langle A (t) u(t), u_m (t)-u(t) \rangle _H \nonumber \\= & {} -{{\mathfrak {R}}}{{\mathfrak {e}}}\langle A (t) u_m (t), u_m (t) - u(t) \rangle _H \nonumber \\&+ {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A (t) u_m (t),{\mathbb {P}} u (t) - u(t) \rangle _H\nonumber \\&+{{\mathfrak {R}}}{{\mathfrak {e}}}\langle A (t) u(t), u_m (t)-u(t) \rangle _H \nonumber \\\le & {} {{\mathfrak {R}}}{{\mathfrak {e}}}\langle A (t) u_m (t),{\mathbb {P}} u (t) - u(t) \rangle _H \nonumber \\= & {} a(t,u_m (t), {\mathbb {P}} u (t) - u(t)) \nonumber \\\le & {} M \Vert u_m (t) \Vert _V \Vert u(t)-{\mathbb {P}} u(t) \Vert _V, \end{aligned}$$
(3.4)

since A is accretive. By integrating, from the last and by Hölder inequality, one gets

$$\begin{aligned} \Vert z_m (t) \Vert _H^2 - \Vert z_m(s) \Vert _H^2\le & {} 2M \Vert u_m \Vert _{L^2 ([s,T],V)} \Vert u-{\mathbb {P}} u \Vert _{L^2 ([s,T],V)}\nonumber \\\le & {} 2M \Vert {\mathbb {P}} x \Vert _H \Vert u-{\mathbb {P}} u \Vert _{L^2 ([s,T],V)}. \end{aligned}$$
(3.5)

Fix \(\varepsilon >0\) and let \(u_\varepsilon \) a continuous \(\varepsilon /4-\) approximation of u in \({L^2 ([s,T],V)}.\) Let \(\tau _\varepsilon \) be such that

$$\begin{aligned} (T-s) \Vert (I - {\mathbb {P}}) u_{\varepsilon } (\tau _\varepsilon )\Vert _V^2 = \int _s^T \Vert (I - {\mathbb {P}}) u_{\varepsilon } (t) \Vert _V^2 dt=\Vert (I- {\mathbb {P}}) u_{\varepsilon } \Vert ^2_{L^2 ([s,T],V)} \end{aligned}$$

and let m be big enough so that \((T-s) \Vert (I - {\mathbb {P}}) u_{\varepsilon } (\tau _\varepsilon )\Vert _V^2<\varepsilon ^2/4.\) We immediately derive

$$\begin{aligned}&\Vert u-{\mathbb {P}} u \Vert _{L^2 ([s,T],V)} \le 2\Vert u - u_\varepsilon \Vert _{L^2 ([s,T],V)} + \Vert (I- {\mathbb {P}}) u_{\varepsilon } \Vert _{L^2 ([s,T],V)} \le \varepsilon . \end{aligned}$$

Note that \(\Vert z_m(s) \Vert _H =\Vert (I-{\mathbb {P}})x\Vert _H \rightarrow 0\) as \(m\rightarrow \infty \) since the embedding of V into H is continuous, hence by (3.5) one gets

$$\begin{aligned} \Vert z_m (t) \Vert _H < 2 \varepsilon \end{aligned}$$

for m large enough and we have proved the lemma. \(\square \)

Remark 3.5

The thesis of previous lemma remains true if \(E_m(t,s)\) is replaced with the adjoint \(E_m(t,s)^*\) and E(t, s) is replaced by \(E(t,s)^*.\) Indeed, the following formula holds for the adjoint (see also [19, 22]):

$$\begin{aligned} E(t,s)^* x = E^r (T-s,T-t) x \text { for any fixed }x\in H \text { and } (t,s)\in \Delta , \end{aligned}$$

where \( \{ E^r (t,s) \} \) is the evolution system associated to the operator \(A^r,\) which in turn is generated by the form \(a^r(t,u,v):=\overline{a(T-t,v,u)},\) which also satisfies properties \((H1)-(H4).\) Here the overline \(\overline{(\cdot )}\) represents the complex conjugate.

Note that, for a fixed \(m\in {\mathbb {N}},\)

$$\begin{aligned} (a_m)^r (t,u,v)= & {} \overline{a_m (T-t, v,u)} \\= & {} \overline{a (T-t,{\mathbb {P}} v,{\mathbb {P}} u)} + \alpha \overline{\langle (I-{\mathbb {P}}) v, (I-{\mathbb {P}}) u \rangle _V} \\= & {} a^r (t,{\mathbb {P}} u,{\mathbb {P}} v) + \alpha \langle (I-{\mathbb {P}}) u, (I-{\mathbb {P}}) v \rangle _V. \end{aligned}$$

This last fact proves that \(E_m(t,s)^* x = E_m^r (T-s,T-t),\) where \(E_m^r (t,s)\) is associated to the form \(a^r (t,{\mathbb {P}} u,{\mathbb {P}} v) + \alpha \langle (I-{\mathbb {P}}) u, (I-{\mathbb {P}}) v \rangle _V.\) Lemma 3.4 can be then applied to \(E^r(t,s)\) and \(E_m^r(t,s)\) to get the result.

4 Nonlinear superposition operators

We will make use of the next lemma, which takes inspiration from [9, Lemma 7.11] (see also [37]). Statement and proof are given in the particular setting of the \(L^2\) space of H-valued functions, though further extensions to a more general setting might be possible. We recall the following

Definition 4.1

Given two Banach spaces X and Y,  a function \(F:X\rightarrow Y\) is said to be demicontinuos if for any sequence \(\{x_n\}_{n\in {\mathbb {N}}} \subset X\) strongly converging to \(x\in X,\) one has that \(\{ F(x_n) \}_{n\in {\mathbb {N}}}\) weakly converges to F(x) or, in other words, that

$$\begin{aligned} w-\lim _{n\rightarrow \infty }F(x_n) = F(x)\text { whenever }x_n\rightarrow x. \end{aligned}$$

Lemma 4.2

Suppose that \(f:[0,T]\times H \rightarrow H\) satisfies

  1. (F1)

    \(f(\cdot , x)\) is measurable for any \(x\in H;\)

  2. (F2)

    \(f(t, \cdot )\) is demicontinuous in H for any fixed \(t\in [0,T].\)

  3. (F3)

    There exist \(a>0\) and \(b \in L^2 ([0,T],{\mathbb {R}}_+)\) such that

    $$\begin{aligned} \Vert f(t,x) \Vert _H \le a \Vert x \Vert _H + b(t). \end{aligned}$$
    (4.1)

Then the superposition operator \(N_f : L^2 ([0,T],H) \rightarrow L^2 ([0,T], H)\) given by \(N_f (u) (t):=f(t,u(t))\) is well-defined and maps bounded sets into bounded sets; moreover, it is demicontinuous.

Proof

The fact that \(N_f\) is well-defined and maps bounded sets into bounded sets is given by Theorem 1.7. It remains to prove that the superposition operator is demicontinuous.

Hence, let \(\{u_m\}\subset L^2 ([0,T],H)\) be a sequence which converges strongly to u,  then for a.e. \(t\in [0,T]\) it holds \(u_m(t) \rightarrow u(t)\) in H;  on the other hand, \(f(t,u_m(t))\rightharpoonup f(t,u(t))\) follows by demicontinuity assumption on f.

On the hand, we observe that \(\{N_f (u_m) \}\) is bounded in \(L^2([0,T],H)\) and, by reflexivity, we can assume it converges weakly to \(v\in L^2([0,T],H)\) up to subsequences; thus, it remains to prove that \(v(t)=f(t,u(t)).\)

Fix a countable and dense sequence \(\{ e_j \}\) in H;  then for any \(j \in {\mathbb {N}}\)

$$\begin{aligned} \langle f(t, u_m(t)), e_j \rangle _H \rightarrow \langle f(t, u(t)), e_j \rangle _H \text{ as } m\rightarrow \infty \end{aligned}$$

for a.e. \(t\in [0,T]\) and by Egorov Theorem there exists a null set \(A_j\) such that the convergence is uniform on \([0,T]\setminus A_j.\) Since \(\bigcup _{j\in {\mathbb {N}}} A_j\) is also null, it holds for any \(j\in {\mathbb {N}}\) that

$$\begin{aligned} \int _0^T | \langle f(t,u_m (t)) - f(t,u(t)), e_j \rangle _H | dt \rightarrow 0 \text{ as } m\rightarrow \infty . \end{aligned}$$
(4.2)

Recalling the fact that \(\{N_f (u_m) \}\) weakly converges to v in \(L^2([0,T],H),\) one in particular gets

$$\begin{aligned} \lim _{m\rightarrow \infty } \int _0^T | \langle N_f(u_m)(t) - v(t), e_j \rangle _H | dt = 0. \end{aligned}$$

The latter together with (4.2) bring to \(\langle f(t,u(t))-v(t), e_j \rangle _H = 0\) for any \(j\in {\mathbb {N}}\) and by density \(v=N_f(u)\) follows. \(\square \)

It is worth noting that the class of demicontinuous function is wide and also includes maximally monotone operators, as we point out in Sect. 6.

5 Existence and regularity of solutions

This section is devoted to the study on the existence of solution in \(H^1 ([0,T],H) \cap L^2([0,T],V)\) of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A(t)u(t)=f(t,u(t)) \qquad t\in [0,T]\ \\ u(0)=g(u), \end{array}\right. } \end{aligned}$$
(5.1)

where

  1. (i)

    V and H are Hilbert spaces, with V densely embedded into H and the embedding \(V\hookrightarrow H\) is compact.

  2. (ii)

    \(\{A(t): t\in [0,T]\}\) is generated by a sesquilinear form a which satisfies (H1)–(H4) and (S).

  3. (iii)

    \(f:[0,T]\times H \rightarrow H\) satisfies conditions (F1)–(F3) in Lemma 4.2 and it holds the following (T) There exists two real numbers \(R_{0}>r_{0}>0\) such that for any \(x\in H\) with \(\Vert x\Vert _H \in (r_{0},R_{0})\)

    $$\begin{aligned} {{\mathfrak {R}}}{{\mathfrak {e}}}( \langle f(t,x),x \rangle _H ) \le 0\qquad \forall t\in [0,T]. \end{aligned}$$
    (5.2)
  4. (iv)

    \(g: L^2([0,T],H)\rightarrow V\) is demicontinuous, maps bounded sets into bounded sets and the condition

    $$\begin{aligned} \Vert g(u) \Vert _H < r \text { whenever }\frac{\Vert u \Vert _{L^2([0,T],H)}}{\sqrt{T}} =:r \in (r_0 , R_0 ) \end{aligned}$$
    (5.3)

    holds, where \(r_0\) and \(R_0\) are the constants from condition (T).

Theorem 5.1

Suppose that the above conditions hold, then Problem (1.75.1) has a solution \(u_* \in H^1 ([0,T],H) \cap L^2([0,T],V);\) moreover, the estimate

$$\begin{aligned}&\Vert u_* \Vert _{H^1 ([0,T],H)} + \Vert u_* \Vert _{L^2([0,T],V)} + \Vert A(\cdot ) u_*(\cdot ) \Vert _{L^2([0,T],H)}\nonumber \\&\quad \le C_1 \left( 2\max \{ar_0\sqrt{T}, \Vert b\Vert _{L^2([0,T],{\mathbb {R}}_+)}\} + g^* \right) \end{aligned}$$
(5.4)

holds, where \(g^* :=\sup \{\Vert g(u)\Vert _V:\Vert u\Vert _{L^2([0,T],H)}\le r_0\sqrt{T}\}\) and \(C_1>0\) is the constant from (1.5).

Proof

Let \(m\in {\mathbb {N}}\) and \(R\in (r_0\sqrt{T},R_0\sqrt{T});\) we will denote by \(\bar{B}_{R,m}\subset L^2([0,T],H_m)\) the set

$$\begin{aligned} \bar{B}_{R,m}:=\left\{ u\in L^2([0,T],H_m): \Vert u(\cdot )\Vert _{L^2([0,T],H)}\le R\right\} \end{aligned}$$

and let \(A_m(t)\) be the operator given in (3.1).

For a fixed \(w \in \bar{B}_{R,m},\) we will at first consider the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A_m(t)u(t)= {\mathbb {P}} f(t,w(t)), \quad a.e. \quad t\in [0,T]\ \\ u(0)={\mathbb {P}}_mg(w). \end{array}\right. } \end{aligned}$$
(5.5)

From Lemmas 2.1-(iv), 3.3 and 4.2 that (5.5) has a unique solution in \(H^1([0,T],H_m)\cap L^2([0,T],V)\cap C([0,T],H_m)\) which can be represented by

$$\begin{aligned} u_w(t):=E_m(t,0){\mathbb {P}}_m g(w) +\int _0^t E_m(t,s) {\mathbb {P}} N_f(w(s)) ds \qquad t\in [0,T], \end{aligned}$$

where \(\{E_m(t,s)\}_{(t,s)\in \Delta }\) is the evolution system associated to \(A_m(t)\) and \(N_f\) is the superposition operator \(N_f (w)(t):=f(t,w(t)).\) Also note that the estimate

$$\begin{aligned} \Vert u_w(t)\Vert _{H^1([0,T],H)} \le C \left( \Vert {\mathbb {P}} g(w) \Vert _V + \Vert N_f w \Vert _{L^2([0,T],H)}\right) \end{aligned}$$

holds true for some constant \(C>0\) (see Theorem 1.3).

The next step is to define the map \(S:[0,1]\times \bar{B}_{R,m}\rightarrow L^2([0,T],H_m)\) by

$$\begin{aligned} S(\lambda ,w(t))= \lambda E_m(t,0){\mathbb {P}}_m g(w) +\int _0^t \lambda E_m(t,s) {\mathbb {P}} N_f(w(s)) \qquad t\in [0,T]. \end{aligned}$$
(5.6)

and to prove that S satisfies the hypotheses of Theorem 1.8.

Firstly observe that \(S(0,x)=0\in {\text {int}}(\bar{B}_{R,m});\) about the complete continuity of S, firstly let \(\{(\lambda _k,w_k)\}_{k\in {\mathbb {N}}}\subset [0,1]\times \bar{B}_{R,m}\) be a sequence such that \(\lambda _k\rightarrow \lambda _0\) and \(w_k\rightarrow w_0.\)

We show that \(S(\lambda _k,w_k) \rightarrow S(\lambda _0,w_0)\) in \(L^2([0,T],H)\), as \(k\rightarrow \infty \). It is not difficult to note that, for \(t\in [0,T]\),

$$\begin{aligned}&\Vert \lambda _k E_m(t,0){\mathbb {P}} g(w_k)-\lambda _0E_m(t,0){\mathbb {P}} g(w_0)\Vert _{H}\\&\quad \le |\lambda _k-\lambda _0|\Vert E_m(t,0){\mathbb {P}} g(w_0)\Vert _{H}+\lambda _k\Vert E_m(t,0){\mathbb {P}} (g(w_k)-g(w_0))\Vert _{H}. \end{aligned}$$

The latter implies that

$$\begin{aligned}&\Vert \lambda _k E_m(\cdot ,0){\mathbb {P}} g(w_k)-\lambda _0E_m(\cdot ,0){\mathbb {P}} g(w_0)\Vert _{L^2([0,T],H_m)} \\&\quad \le C( |\lambda _k-\lambda _0| \Vert g(w_0)\Vert _H + \Vert {\mathbb {P}} g(w_k) - {\mathbb {P}} g(w_0)\Vert _H ). \end{aligned}$$

Since \({\mathbb {P}}\) is weak-to-strong continuous, while g is demicontinuous, we derive that \({\mathbb {P}} g (w_{k}) \rightarrow {\mathbb {P}} g (w_{0}) \), as \(k\rightarrow \infty .\) We have then proved that

$$\begin{aligned} \lim _{k\rightarrow \infty } \lambda _k E_m(\cdot ,0){\mathbb {P}} g(w_k) = \lambda _0 E_m(\cdot ,0){\mathbb {P}} g(w_0) \text { in } L^2([0,T],H_m). \end{aligned}$$
(5.7)

On the other hand, since \({\mathbb {P}}\) is weak-to-strong continuous, while \(N_f\) is demicontinuous by Lemma 4.2, we derive that \({\mathbb {P}} N_f (w_{k}) (s) \rightarrow {\mathbb {P}} N_f (w_{0}) (s)\), as \(k\rightarrow \infty .\) Note that \(w_k (s) \rightarrow w_0 (s)\) a.e. uniformly on [0, T] by Egorov theorem, so that \(\Vert w_k (s) \Vert _H \le \Vert w_0 (s) \Vert _H+\varepsilon _0\) holds uniformly on [0, T] for some \(\varepsilon _0 >0.\) From condition (F3) and the contractivity of the evolution system, it is derived that

$$\begin{aligned}&\Vert E_m(t,s){\mathbb {P}} N_f (w_{k}) (s)\Vert _{H} \nonumber \\&\quad \le a \Vert w_{0} (s) \Vert _{H} + \varepsilon _0 +b (s) \text{ for } \text{ a.e. } t\in [0,T], \end{aligned}$$
(5.8)

with

$$\begin{aligned} \Big \Vert a \Vert w_0 (\cdot ) \Vert _{H} + \varepsilon _0 + b(\cdot )\Big \Vert _{L^1([0,T],{\mathbb {R}}_+)} \le C \left( R+\varepsilon _0 +\Vert b\Vert _{L^2([0,T],{\mathbb {R}}_+)} \right) . \end{aligned}$$

Then, by Lebesgue’s Dominated Convergence Theorem,

$$\begin{aligned} \int _0^t E_m(t,s){\mathbb {P}}_mN_f (w_{k}) (s)ds\rightarrow \int _0^t E_m(t,s){\mathbb {P}}_mN_f (w_{0}) (s)ds \end{aligned}$$

a.e. on \(t\in [0,T]\) as \(k\rightarrow \infty ;\) since \(\lambda _k\rightarrow \lambda _0\) too, then

$$\begin{aligned} \lim _{k\rightarrow \infty } \lambda _k \int _0^t E_m(t,s){\mathbb {P}}_mN_f (w_{k}) (s)ds = \lambda _0 \int _0^t E_m(t,s){\mathbb {P}}_mN_f (w_{0}) (s)ds \end{aligned}$$

for a.e. \(t\in [0,T].\) Similarly as in (5.8) it can be proved that the above convergence is dominated by the same function in \(L^2([0,T],H_m).\) Hence,

$$\begin{aligned} \int _0^{(\cdot )} E_m(\cdot ,s){\mathbb {P}}_mN_f (w_{k}) (s)ds\rightarrow \int _0^{(\cdot )} E_m(\cdot ,s){\mathbb {P}}_mN_f (w_{0}) (s)ds \end{aligned}$$

in \(L^2([0,T],H)\) as \(k\rightarrow \infty \) and so

$$\begin{aligned} \lim _{k\rightarrow \infty } S(\lambda _k,w_k) = S(\lambda _0,w_0) \text { in }L^2([0,T],H_m) \end{aligned}$$

and the continuity of the operator S is proved.

We next show that S maps bounded sets into compact ones. To this end, it is enough to show that \(S([0,1],\bar{B}_{R,m})\) is relatively compact in \(L^2([0,T],H_m)\). Observe that, whenever \(u\in S([0,1],\bar{B}_{R,m}),\) then it is the unique solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A_m(t)u(t)= {\mathbb {P}} \lambda f(t,w(t)), \quad a.e. \quad t\in [0,T]\ \\ u(0)={\mathbb {P}}_m \lambda g(w) \end{array}\right. } \end{aligned}$$
(5.9)

for some \(\lambda \in [0,1]\) and \(w\in \bar{B}_{R,m}.\) By Lemma 3.3, (F3) and Theorem 1.3, it is true that \(u\in H^1([0,T],H_m)\) and that

$$\begin{aligned} \Vert u \Vert _{H^1([0,T],H_m)}\le & {} C_1 \left( \Vert {\mathbb {P}} g(w) \Vert _V + a \Vert w \Vert _{L^2([0,T],V_m)} + \Vert b\Vert _{L^2([0,T],{\mathbb {R}}_+)} \right) \nonumber \\\le & {} C_1 \left( \sup \{ \Vert g(w)\Vert _V:\Vert w\Vert _{L^2([0,T],H)}\le R\} + 1+ a R + \Vert b \Vert _{L^2([0,T],{\mathbb {R}}_+)} \right) \nonumber \\< & {} \infty \end{aligned}$$
(5.10)

since g maps bounded sets into bounded sets and where we supposed for simplicity that \(\Vert g(u)-{\mathbb {P}} g(u)\Vert _V<1.\)

This last fact shows that \(S([0,1],\bar{B}_{R,m})\subset H^1([0,T],H_m)\) is uniformly bounded and hence relatively compact in \(L^2([0,T],H_m)\), by Theorem 1.9. In order to apply Theorem 1.8, it remains to prove that the set \(\{ u\in \bar{B}_{R,m}: S(\lambda , u)=u \text{ for } \text{ some } \lambda \in [0,1] \}\) is bounded and has no intersection with the boundary of \(\bar{B}_{R,m}.\) To this end, fix \(\lambda \in (0,1)\) and suppose that \({\tilde{u}}=S(\lambda , {\tilde{u}})\) for some \({\tilde{u}}\) with \(\Vert {\tilde{u}} \Vert _{L^2([0,T],H_m)}=R\in (r_0 \sqrt{T}, R_0 \sqrt{T}).\) Note that (5.3) implies that

$$\begin{aligned} \Vert {\tilde{u}}(0) \Vert _H^2\le \Vert g({\tilde{u}})\Vert ^2_H< \frac{R^2}{{T}} = \frac{1}{{T}} \Vert {\tilde{u}} \Vert ^2_{L^2([0,T],H_m)}. \end{aligned}$$

Since \({\tilde{u}}\) is continuous and by the mean value theorem, two points \(t_1, t_2 \in (0,T]\) must exist such that

$$\begin{aligned} r_0^2< \Vert {\tilde{u}}(t_1) \Vert ^2_{H}< \Vert {\tilde{u}}(t_2 ) \Vert ^2_{H}= \frac{1}{{T}} \Vert {\tilde{u}}\Vert ^2_{L^2([0,T],H_m)}<R_0^2. \end{aligned}$$
(5.11)

Note that condition (T) implies that for any \(t\in [t_1,t_2]\)

$$\begin{aligned} \langle {\mathbb {P}} N_f ({\tilde{u}}) (t), {\tilde{u}}(t) \rangle _{H} = \langle f(t,{\tilde{u}}(t)), {\mathbb {P}} {\tilde{u}}(t) \rangle _{H} = \langle f(t,{\tilde{u}}(t)), {\tilde{u}}(t) \rangle _H\le 0. \end{aligned}$$
(5.12)

Since \(A_m\) is accretive, (5.11) holds and by (5.12), the contradiction

$$\begin{aligned} 0< & {} \frac{ \Vert {\tilde{u}} (t_2) \Vert _{H}^2 - \Vert {\tilde{u}} (t_1) \Vert _{H}^2 }{2} =\int _{t_1}^{t_2} \frac{d}{dt} \left( \frac{1}{2} \Vert {\tilde{u}}(t) \Vert ^2\right) dt\nonumber \\= & {} \int _{t_1}^{t_2} {{\mathfrak {R}}}{{\mathfrak {e}}}(\langle {\tilde{u}}'(t), {\tilde{u}}(t) \rangle _H) dt = \int _{t_1}^{t_2} - {{\mathfrak {R}}}{{\mathfrak {e}}}(\langle A_m(t){\tilde{u}}(t), {\tilde{u}}(t) \rangle _H)dt \nonumber \\&+\lambda \int _{t_1}^{t_2} {{\mathfrak {R}}}{{\mathfrak {e}}}( \langle {\mathbb {P}} N_f ({\tilde{u}}) (t), {\tilde{u}}(t) \rangle _H )dt \le 0 \end{aligned}$$
(5.13)

arises. We have then proved that \(\Vert {\tilde{u}} \Vert _{L^2([0,T],H_m)}<R\) and then \(S(1,\cdot )\) has a fixed point by Theorem 1.8, that is there exists a solution \(u_m\) to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A_m(t)u(t)= {\mathbb {P}} f(t,u(t)), \quad a.e. \quad t\in [0,T]\ \\ u(0)={\mathbb {P}}_mg(u); \end{array}\right. } \end{aligned}$$
(5.14)

moreover \(u_m \in H^1([0,T],H_m)\cap L^2([0,T],V)\cap C([0,T],H).\) Consider the sequence \(\{ u_m \}_{m\in {\mathbb {N}}} \subset L^2([0,T],H);\) then by Remark 1.4 and by reasoning as in (5.10) it follows that

$$\begin{aligned} \Vert u \Vert _{H^1([0,T],H)} + \Vert u \Vert _{L^2([0,T],V)} \le C < + \infty \end{aligned}$$

for some constant C not depending on \(m\in {\mathbb {N}}\) or, in other words, that \(\{ u_m \}_{m\in {\mathbb {N}}}\) is bounded in \(H^1([0,T],H)\cap L^2([0,T],V).\) By Theorem 1.9, the latter space embeds compactly in \(L^2([0,T],H)\) since V is also compactly embedded onto H. In particular, \(\{ u_m \}_{m\in {\mathbb {N}}}\) is relatively compact in \(L^2([0,T],H),\) that is a subsequence \(\{u_{m_k}\}_{k\in {\mathbb {N}}}\) exists which converges to a point \(u_*\in L^2([0,T],H).\) We may also assume that \(u_{m_k}(t)\rightarrow u_*(t)\) as \(k\rightarrow \infty \) for a.e.    \(t\in [0,T].\)

By passing to a further subsequence, it can be seen that for any fixed \(t\in [0,T]\)

$$\begin{aligned} \int _0^t E_{m_k}(t,s){\mathbb {P}}_{m_k} f(s,u_{m_k}(s)) - E(t,s) f(s,u_*(s))ds \rightarrow 0. \end{aligned}$$
(5.15)

Indeed, for a fixed \(t\in [0,T],\) consider

$$\begin{aligned} y_{m_k}(t):= \int _0^t E_{m_k}(t,s){\mathbb {P}}_{m_k} f(s,u_{m_k}(s))- E(t,s)f(s,u_*(s)) ds \end{aligned}$$

and note that \(\{ y_{m_k} \}_{k\in {\mathbb {N}}} \) can be seen as a bounded sequence in \(H^1 ([0,T],H)\) by Lemma 2.1-(iv) and since \(H^1 ([0,T],H_{m_k})\hookrightarrow H^1 ([0,T],H).\) Reasoning as before and passing to further subsequences if necessary, it is readily proved that \(y_{m_k}(t)\rightarrow y_*(t)\) as \(k\rightarrow \infty .\) On the other hand, fix \(x\in H\) and \((t,s)\in \Delta \) note that,

$$\begin{aligned}&\langle E_{m_k}(t,s){\mathbb {P}}_{m_k} f(s,u_{m_k}(s)) - E(t,s) f(s,u_*(s)), x \rangle _H \nonumber \\&\quad =\langle E_{m_k}(t,s){\mathbb {P}}_{m_k} ( f(s,u_{m_k}(s)) - f(s,u_*(s))), x \rangle _H\nonumber \\&\quad \quad + \langle ( E_{m_k}(t,s){\mathbb {P}}_{m_k} - E(t,s) )f(s,u_*(s)),x \rangle _H\nonumber \\&\quad =\langle f(s,u_{m_k}(s)) - f(s,u_*(s)), E(t,s)^* x \rangle _H \nonumber \\&\quad \quad + \langle f(s,u_{m_k}(s)) - f(s,u_*(s)) ,(E_{m_k}(t,s){\mathbb {P}}_{m_k})^* x - E(t,s)^* x \rangle _H\nonumber \\&\quad \quad + \langle ( E_{m_k}(t,s){\mathbb {P}}_{m_k} - E(t,s) )f(s,u_*(s)),x \rangle _H. \end{aligned}$$
(5.16)

Let \(k\rightarrow \infty \) and observe that

$$\begin{aligned} \langle f(s,u_{m_k}(s)) - f(s,u_*(s)), E(t,s)^* x \rangle _H \rightarrow 0 \end{aligned}$$

since \(f(s,\cdot )\) is demicontinuous, while

$$\begin{aligned} \langle f(s,u_{m_k}(s)) - f(s,u_*(s)) ,(E_{m_k}(t,s){\mathbb {P}}_{m_k})^* x - E(t,s)^* x \rangle _H\rightarrow 0 \end{aligned}$$

by Remark 3.5 and since \( f(s,u_{m_k}(s)) - f(s,u_*(s))\) is bounded; lastly,

$$\begin{aligned} \langle ( E_{m_k}(t,s){\mathbb {P}}_{m_k} - E(t,s) )f(s,u_*(s)),x \rangle _H\rightarrow 0 \end{aligned}$$

by Lemma 3.4.

Then (5.16) implies that

$$\begin{aligned} \lim _{k\rightarrow \infty } \langle E_{m_k}(t,s){\mathbb {P}}_{m_k} f(s,u_{m_k}(s)) - E(t,s) f(s,u_*(s)), x \rangle _H = 0. \end{aligned}$$

Also, it is easily seen that for any fixed \(t\in [0,T]\) and any \(s\in [0,t],\)

$$\begin{aligned} | \langle E_{m_k}(t,s){\mathbb {P}}_{m_k} f(s,u_{m_k}(s)) - E(t,s) f(s,u_*(s)), x \rangle _H |\le & {} \Vert x\Vert +(1+a^2)\Vert x\Vert ^2\\&+a^2 \Vert u(s)\Vert _H^2+|b(s)|^2 \end{aligned}$$

where we have supposed that \(\Vert u_{m_k}(s) \Vert _H \le 1+\Vert u_*(s)\Vert _H\) a.e. uniformly on [0, t] and for k large enough. Since the last term of the inequality lies in \(L^1([0,t],H),\) the Lebesgue’s Dominated Convergence Theorem implies that

$$\begin{aligned} \langle y_{m_k} (t), x \rangle _H= & {} \left\langle \int _0^t E_{m_k}(t,s){\mathbb {P}}_{m_k} f(s,u_{m_k}(s))- E(t,s)f(s,u_*(s)) ds , x \right\rangle _H\\= & {} \int _0^t \langle E_{m_k}(t,s){\mathbb {P}}_{m_k} f(s,u_{m_k}(s))- E(t,s)f(s,u_*(s)),x\rangle _H ds \rightarrow 0 \text { as }k\rightarrow \infty , \end{aligned}$$

where we have used [48, Proposition 23.9]. By uniqueness of the weak limit and since \(x\in H\) is arbitrary, it follows that \(y_* = 0\) and hence (5.15) is proved.

Further we note that

$$\begin{aligned} \Vert E_{m_k}(t,0) {\mathbb {P}}_{m_k} g(u_{m_k}) - E(t,0)g(u_*)\Vert _H\le & {} \Vert E_{m_k}(t,0) {\mathbb {P}}_{m_k}(g(u_{m_k})-g(u_*))\Vert _H \\&+ \Vert (E_{m_k}(t,0){\mathbb {P}}_{m_k} - E(t,0))g(u_*)\Vert _H \end{aligned}$$

and by Lemma 3.4, \(\Vert (E_{m_k} (t,0) {\mathbb {P}}_{m_k}- E(t,0))g(u_*)\Vert _H\rightarrow 0\) as \(k\rightarrow \infty \). In order to prove that \(\Vert E_{m_k}(t,0) {\mathbb {P}}_{m_k}(g(u_{m_k})-g(u_*))\Vert _H\rightarrow 0\), as above let us introduce

$$\begin{aligned} z_{m_k}(t):= E_{m_k}(t,0) {\mathbb {P}}_{m_k}(g(u_{m_k})-g(u_*)) \end{aligned}$$

that is a bounded sequence in \(H^1 ([0,T],H)\). Passing to further subsequences if necessary, it is readily proved that \(z_{m_k}(t)\rightarrow z_*(t)\) as \(k\rightarrow \infty .\) On the other hand, fix \(x\in H\) and \((t,s)\in \Delta \) and note that,

$$\begin{aligned} \left\langle z_{m_k}, x \right\rangle _H= & {} \langle g(u_{m_k})-g(u_*), (E_{m_k}(t,0) {\mathbb {P}}_{m_k})^* x \rangle _H \\= & {} \langle g(u_{m_k})-g(u_*),(E_{m_k}(t,0){\mathbb {P}}_{m_k})^* x- E(t,0)^* x \rangle _H\\&+ \langle g(u_{m_k})-g(u_*),E(t,0)^*x \rangle _H. \end{aligned}$$

When \(k\rightarrow \infty \), we observe that \(\langle g(u_{m_k})-g(u_*),E(t,0)^*x \rangle _H \rightarrow 0\) since \(g(\cdot )\) is demicontinuous, while

$$\begin{aligned} \left\langle g(u_{m_k})-g(u_*), \left( E_{m_k}(t,0){\mathbb {P}}_{m_k}\right) ^* x- E(t,0)^* x \right\rangle _H\rightarrow 0 \end{aligned}$$

by Remark 3.5 and since \(g(u_{m_k})-g(u_*)\) is bounded. This proves that \(\langle z_{m_k}(t),x\rangle _H \rightarrow 0\), as \(k\rightarrow \infty \) and so \(z_*=0\).

The fact that \(g(u_{m_k})\rightarrow g(u_*)\) as \(k\rightarrow \infty \) follows by arguing as before. Putting all together, we have then proved that for any \(t\in [0,T],\)

$$\begin{aligned} u_* (t)= & {} \lim _{k\rightarrow \infty } u_{m_k} (t)\\= & {} \lim _{k\rightarrow \infty } E_{m_k}(t,0){\mathbb {P}}_{m_k} g(u_{m_k}) + \int _0^t E_{m_k}(t,s){\mathbb {P}} f(s,u_{m_k}(s))ds \\= & {} E(t,0)g(u_*) + \int _0^t E(t,s)f(s,u_*(s))ds. \end{aligned}$$

Hence, by Lemma 2.1, \(u_*\in H^1 ([0,T],H) \cap L^2([0,T],V) \) solves Problem (1.75.1). Let \(g^*_R :=\sup \{\Vert g(u)\Vert _V:\Vert u\Vert _{L^2([0,T],H)}\le R \sqrt{T}\}\); it holds

$$\begin{aligned}&\Vert u_* \Vert _{H^1 ([0,T],H)} + \Vert u_* \Vert _{L^2([0,T],V)} + \Vert A(\cdot ) u_*(\cdot ) \Vert _{L^2([0,T],H)} \\&\quad \le C_1 \left( \Vert N_f (u_*) \Vert _{L^2([0,T],H)} + \Vert g(u_*) \Vert _V \right) \\&\quad \le C_1 \left( 2\max \left\{ a\Vert u_*\Vert _{L^2([0,T],H)}, \Vert b\Vert _{L^2([0,T],{\mathbb {R}}_+)} \right\} + g^*_R \right) \\&\quad \le C_1 \left( 2\max \left\{ aR, \Vert b\Vert _{L^2([0,T],{\mathbb {R}}_+)} \right\} + g^*_R \right) . \end{aligned}$$

To reach the estimate (5.4), we observe that in the latter inequality, R can be chosen to be arbitrarily close to \(r_0.\) \(\square \)

Corollary 5.2

Let \(\{A(t): t\in [0,T]\}\) be generated by a sesquilinear form \(a(\cdot ,\cdot ,\cdot )\) which satisfies (H1)–(H3) and (H4\(^*\)). Moreover, suppose that (S) holds and that \(f[0,T]\times H \rightarrow H\) satisfies conditions (F1)–(F3) in Lemma 4.2 for some \(b\in L^\infty ([0,T],{\mathbb {R}}_+)\). Suppose also that g is demicontinuous and such that \( g_*:=\sup \{ \Vert g(u) \Vert _V : u\in L^2([0,T],H)\}<+\infty \) holds. If V is compactly embedded in H, then problem

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A(t)u(t)= f(t,u(t)), \quad a.e. \quad t\in [0,T]\ \\ u(0)= g(u) \end{array}\right. } \end{aligned}$$
(5.17)

admits a solution in \(H^1 ([0,T],H) \cap L^2([0,T],V)\)

Proof

We have already seen in Remark 1.5 that by setting \(v(t):=e^{-\mu t} u(t)\), problem (5.17) is equivalently rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} v'(t)+(A(t)+\mu I)v(t)= e^{-\mu t} f(t,e^{\mu t}v(t)), \quad a.e. \quad t\in [0,T]\ \\ v(0)= g ( e^{\mu \cdot } v(\cdot ) ). \end{array}\right. } \end{aligned}$$
(5.18)

We will see that the same trick permits us to weaken the hypothesis on the nonlinear term. Indeed, choose \(\mu := \delta + \epsilon ,\) where \(\epsilon > a\), being a the constant from (F3). We rewrite (5.18) as

$$\begin{aligned} {\left\{ \begin{array}{ll} v'(t)+{{\hat{A}}} (t)v(t)= {\hat{f}}(t,v(t)), \quad a.e. \quad t\in [0,T]\ \\ v(0)= {\hat{g}}(v), \end{array}\right. } \end{aligned}$$
(5.19)

where \({{\hat{A}}}(t) = A(t) + \delta I\) is accretive, \({\hat{g}}:v\in L^2([0,T],H) \mapsto g( e^{\mu \cdot } v(\cdot ))\in V\) is demicontinuous and bounded and \({\hat{f}}(t,x):=e^{-\mu t} f(t,e^{\mu t}x) - \epsilon x\) still satisfies properties (F1)–(F3). To see that \({\hat{f}}\) also satisfies the transversality condition (T), note that for any \((t,x)\in [0,T]\times H\)

$$\begin{aligned} {{\mathfrak {R}}}{{\mathfrak {e}}}\langle {\hat{f}} (t,x),x \rangle _H= & {} {{\mathfrak {R}}}{{\mathfrak {e}}}\langle e^{-\mu t} f(t,e^{\mu t}x),x \rangle _H -\epsilon \Vert x\Vert ^2 \\\le & {} a \Vert x \Vert ^2 + e^{-\mu t} b(t) \Vert x\Vert -\epsilon \Vert x\Vert ^2\\\le & {} \left( (a-\epsilon )\Vert x\Vert + \Vert b \Vert _\infty \right) \Vert x\Vert . \end{aligned}$$

Since \(\epsilon > a,\) for any \(\Vert x\Vert > \Vert b \Vert _\infty /(\epsilon - a)\) it holds

$$\begin{aligned} {{\mathfrak {R}}}{{\mathfrak {e}}}\langle {\hat{f}} (t,x),x \rangle _H \le 0. \end{aligned}$$

In particular, condition (T) is satisfied for any \(r_0> \frac{\Vert b\Vert _\infty }{\epsilon -a}.\)

Set \(r_0 > \max \left\{ \sqrt{T}g_*, \frac{\Vert b\Vert _\infty }{\epsilon -a }\right\} \) and \(R_0 := +\infty ,\) then \({\hat{g}}\) satisfies the hypotheses of previous result, which can be followed closely to prove the existence of a solution to (5.19). \(\square \)

6 Applications

Models of non-autonomous operators A(t) governed by forms satisfying assumptions (H1)–(H4) and (S) had been widely investigated in the literature (see e.g., [2, 27]). Here we briefly introduce the topic.

Let \(\Omega \) be an open connected and bounded subset of \({\mathbb {R}}^n\) whose boundary is a finite union of parts of rotated graphs of Lipschitz maps, i.e., a strongly Lipschitz domain. Let H be \(L^2(\Omega )\) with the classical Lebesgue measure, while with \(H^1(\Omega )\) we denote the classical Sobolev space consisting of all square-integrable functions \(u:\Omega \rightarrow {\mathbb {R}}\) such that \(\nabla u\) exists in the weak sense and belongs to \(L^2(\Omega ).\) We denote by \(H_0^1(\Omega )\) the closure of \(C_0^\infty (\Omega )\) in \(H^1(\Omega ).\) Since \(\Omega \) is open and bounded, by using Poincaré inequality, we equip \(H_0^1(\Omega )\) with the equivalent norm \(\Vert u\Vert _{H_0^1(\Omega )}:=\Vert \nabla u\Vert _{L_2(\Omega )}\). In particular, by Rellich–Kondrakov’s Theorem, \(H_0^1 (\Omega )\hookrightarrow L^2(\Omega )\) compactly.

Note that H is reflexive and separable, so that \(L^2([0,T],H)\) is also reflexive and separable; moreover, \(H^1([0,T],H)\) is compactly embedded on \(L^2([0,T],H)\).

We recall the following result proved [7], which provides a wide class of operators satisfying condition (S).

Theorem 6.1

Let \(\Omega \subset {\mathbb {R}}^n\) be a strongly Lipschitz domain and A a bounded uniformly elliptic complex matrix on \(\Omega \). Let \(L:=-{\text {div}}(A\nabla ),\) with \(L:V \rightarrow H^1(\Omega )\) and \(V=H^1_0(\Omega )\) or \(V=H^1(\Omega ) .\) Then the domain of the maximal accretive square root \(L^\frac{1}{2}\) agrees with V with equivalence of norms.

To provide a model for sesquilinear forms, we set \(V:=H^{1}(\Omega )\) and define

$$\begin{aligned} a(t,u,v) :=\int _\Omega \sum _{i,j=1}^{n} a_{i,j}(t,x)\partial _iu \ \overline{\partial _jv} \ {\text {d}}x \end{aligned}$$
(6.1)

then \(a:[0,T]\times V \times V \rightarrow {\mathbb {C}}.\) The assumptions on a are the following:

  1. (a)

    Let \(a_{i,j}(\cdot ,\cdot )\in L^\infty ([0,T]\times \Omega ,{\mathbb {R}})\) (\(i,j=1,\dots ,n\)) satisfying:

  2. (a1)

    The (uniform) ellipticity condition

    $$\begin{aligned} \sum _{i,j=1}^{n} a_{i,j}(t,x)\xi _i\xi _j \ge \nu \Vert \xi \Vert ^2, \qquad \forall \xi \in {\mathbb {R}}^n, a.e. \quad (t,x)\in [0,T]\times \Omega , \end{aligned}$$

    where \(\nu >0\) holds.

  3. (a2)

    There exists \(K>0\) such that for any i, j,

    $$\begin{aligned} |a_{i,j}(t_1,\xi )-a_{i,j}(t_2,\xi )|\le K |t_1-t_2|^{\alpha } \end{aligned}$$

    a.e. \(\xi \in \Omega \), \(t_1,t_2\in [0,T]\) and \(\displaystyle \alpha >\frac{1}{2}\)

Proposition 6.2

The sesquilinear form a(t, u, v) satisfies assumptions (H1–H4) and (S).

Proof

While (H1) is directly obtained, (H2) and (H3) derives from the following energy estimate (see [7]) which holds in our setting:

$$\begin{aligned} \alpha \Vert u|^2_{V}\le a(t,u,u) \end{aligned}$$

for \(u\in V\), a.e. in \(t\in [0,T]\) and

$$\begin{aligned} |a(t,u,v)|\le M \Vert u\Vert _V \Vert v\Vert _{V} \end{aligned}$$
(6.2)

for some \(M,\alpha >0\) and \(u,v\in V.\) At the end, note that (H4) follows for \(\omega (t)=Kt^\alpha \) by (a2).

Following [27], define the linear operator \({\mathcal {A}}(t):V\rightarrow V'\) such that \(\langle {\mathcal {A}}(t)u,v\rangle _V=a(t,u,v)\), where \(\langle \cdot ,\cdot \rangle _V\) denotes the usual pairing in \(V'\times V\). For each \(t\in [0,T]\), the part of \({\mathcal {A}}(t)\) on H is given by

$$\begin{aligned} A(t):=-\sum _{i,j=1}^{n}\partial _i\left( a_{i,j}(t,\cdot )\partial _j\right) \end{aligned}$$

on \({\mathcal {D}}(A(t)):=\{u\in V:{\mathcal {A}}(t)u\in H\}\). Fix \(t\in [0,T].\) Thanks to uniform quasi-coercivity and uniform boundedness of \(a(t,\cdot ,\cdot )\), by Theorem 6.1, the square root property holds, i.e., \({\mathcal {D}}(A(t))^{\frac{1}{2}})=V.\) \(\square \)

To better illustrate the application range of our main results, in the sequel we analyze two settings in which Theorem 5.1 applies. The first one regards evolution variational inequality problems. This problem had been widely investigated in the past with a complete understanding when the problem is governed by a maximal monotone operator ([16]). Nevertheless, it represents an interesting test-bed for our result: indeed, we show that hypotheses (F1)–(F3) on the nonlinear term are naturally satisfied by maximal monotone operators with a sublinear growth, so that existence of solutions and regularity properties derive from Theorem 5.1. Lastly, we focus our attention on the nonlocal initial condition \(u(0)=g(u).\) Nonlocal initial boundary value problems for semilinear equations often arise in concrete physical model and, in particular, in heat conduction or diffusion processes ([10]). In particular, multipoint initial conditions are used to describe the diffusion phenomenon of a small amount of gas in a transparent tube, where several consecutive measurements are more effective ([20]).

6.1 Evolution variational inequality problems

As highlighted before, an important class of evolutionary problems is represented by evolution variational inequalities. Evolution variational inequalities have been successfully applied in several fields of science with applications to oligopolistic markets, urban transportation networks, traffic networks, international trade, agricultural and energy markets (see e.g., [21, 23, 25, 38] and references therein). Let \(\varphi : L^2(\Omega ) \rightarrow {\mathbb {R}}\) be a proper, convex and Gateaux differentiable function with a sublinear gradient growth, i.e., there exists \(M,b>0\) such that

$$\begin{aligned} \Vert \nabla \varphi (u) \Vert _{L^2(\Omega )} \le M \Vert u \Vert _{L^2(\Omega )} + b, \end{aligned}$$

for any \(u\in L^2(\Omega ).\)

An example of such functions is given by the class \(C^{1,1}_L\) of the convex and Gateaux differentiable functions with Lipschitz continuous gradient; for more details on the class and its applications to optimization, we refer the reader to [39] and references therein.

Let \(u_0\in H^1(\Omega )\) and A(t) be as in Proposition 6.2; we are interested in the following evolution variational inequality problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \langle u'(t) + A(t)u(t), v - u(t) \rangle _{L^2(\Omega )} \ge \varphi (u(t)) - \varphi (v)\text {, for any } v\in L_2(\Omega )\\ u(0)=u_0. \end{array}\right. } \end{aligned}$$
(6.3)

By following [49, pp. 893–894], we rewrite the previous inequality as

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A(t) u(t) = - \nabla \varphi (u(t)), \\ u(0)=u_0. \end{array}\right. } \end{aligned}$$
(6.4)

In the sequel, we prove that the nonlinearity \(f(t,x):=-\nabla \varphi (x)\) satisfies assumptions (F1)–(F3), thus fullfilling the requests.

Observe that \(\nabla \varphi \) is maximal monotone and has full domain since \(\varphi \) is convex and Gateaux differentiable on \(L^2(\Omega ),\) which implies that \(x\mapsto \nabla \varphi (x)\) is demicontinuous (see [8, Corollary 21.21]) and the same property is then satisfied by \(f(t,\cdot ).\) Condition (F3) immediately derives from the sublinearity of \(\nabla \varphi .\) By setting \(g(u):=u_0,\) we see that the hypotheses of Corollary 5.2 are fulfilled and the following result holds.

Proposition 6.3

Problem 6.3 admits a solution in \(H^1 ([0,T],L^2(\Omega )) \cap L^2([0,T]\), \(H^1(\Omega )).\)

6.2 Semilinear equations with nonlocal initial conditions

As we have already pointed out, several expressions of nonlocal initial conditions had been intensively studied in the literature. Among others, we cite [15] where a multipoint condition had been studied under further compactness assumption, while in [41], the authors considered a general integral nonlocal initial condition of type

$$\begin{aligned} u(0)=\int _{[0,T]} {\mathcal {V}}(u(t)) {\text {d}}t. \end{aligned}$$

In order to better expose the range of applications provided by Theorem 5.1, we will deal with an initial condition of this type.

As highlighted before, existing literature on the topic mainly deals with existence of mild solutions to nonlocal initial value problems. Here, we are interested in proving the existence of strong solutions and further regularities; this result can be achieved by introducing a smoothing term, below represented by the convolution with a sufficiently smooth function.

Let \(\Omega ={\mathbb {R}}^n\) and let A(t) be associated to the sesquilinear form given (6.1); then assumptions (H1)–(H4) and (S) are still valid with \(V=H^1 ({\mathbb {R}}^n)\) (see [27, 42]). We are interested in the following nonlocal evolution problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)+A(t) u(t) = f(t),\ t\in (0,1] \\ \displaystyle u(0)= \int _{I} \varphi *u(t) {\text {d}}t, \end{array}\right. } \end{aligned}$$
(6.5)

where \(f\in L^\infty ([0,1],{\mathbb {R}}),\ I=\bigcup [s_i,t_i] \subset [0,1],\) \(\varphi \in C^1 ({\mathbb {R}}^n)\) is a mollifier with \(\Vert \nabla \varphi \Vert _{L^1({\mathbb {R}}^n)} < 1\) and where \(\varphi * u(t)\) represents the convolution product among the two terms.

Note that by choosing \(\varepsilon >0\) small enough, \(A_\varepsilon (t):=A(t)-\varepsilon I\) still satisfies the required properties, while \(f_\varepsilon (t, x) := f(t)-\varepsilon x\) fulfills requirements \((F1)-(F3)\) and the transversality condition (5.2) for any \(R_0>r_0 > \Vert f \Vert _\infty .\)

By a standard argument, it is promptly derived that \(\displaystyle g(u) = \int _{I} \varphi *u(t) {\text {d}}t\) satisfies

$$\begin{aligned} \Vert g(u) {\Vert ^2}_{H^1({\mathbb {R}}^n)} \le \Vert \nabla \varphi \Vert _{L^1({\mathbb {R}}^n)} \int _{[0,1]} \Vert u(t) \Vert ^2_{L^2({\mathbb {R}}^n)} {\text {d}}t. \end{aligned}$$

The latter proves that \(g: L^2([0,T],{L^2({\mathbb {R}}^n)}) \rightarrow H^1({\mathbb {R}}^n)\) is well-defined, continuous and satisfies (5.3). At this point, an immediate application of Theorem 5.1 brings the following

Proposition 6.4

A solution \(u_* \in H^1 ([0,T],L^2({\mathbb {R}}^n)) \cap L^2([0,T],H^1({\mathbb {R}}^n))\) to problem (6.5) exists for which the a priori estimate (5.4) holds.