Nonautonomous semilinear Wentzell problems in fractal domains

Abstract

We study a nonautonomous semilinear parabolic problem with a dynamical boundary condition in an irregular domain with fractal boundary. Local existence, uniqueness and regularity results for the mild solution are established via a family of evolution operators. A sufficient condition on the initial datum for global existence is given.

1 Introduction

In this paper, we study a nonautonomous semilinear problem in a fractal domain.

We consider the abstract Cauchy problem

$$\begin{aligned} (P)\qquad \left\{ \begin{array}{l} \frac{\textrm{d}u(t)}{\textrm{d}t} = A(t) u(t) + J(u(t)),\qquad 0\le t\le T\\ u(0)=\phi \end{array} \right. \end{aligned}$$

(1.1)

Here, $A(t):\mathcal {D} (A(t))\subset L^2(Q)\rightarrow L^2(Q)$ is the family of operators associated to the energy forms E[t, u] introduced in (3.11)

$$\begin{aligned} E[t,u]= \int _Q|\textrm{D}u|^2\textrm{d}Q+ E_S[u|_S]+ \int _S b(t,P) |u|_S|^2 \, \textrm{d}m, \end{aligned}$$

(1.2)

defined on the domain $[0,T]\times V(Q,S),$

where Q is the bounded open set defined as $Q=\Omega \times I,$ $I=(0,1),$ $\Omega$ is the (open) snowflake domain, $\Omega \subset \mathbb {R}^2$ and $F=\partial \Omega$ is the union of three Koch curves (see Section 1), and $S=F \times I$. Here, $b(t,P):(0,T)\times S\rightarrow \mathbb {R}$ is a function such that

$$\begin{aligned} A) \left\{ \begin{array}{ll} b\in L^\infty ([0,T]\times S), &{} \hbox {} \\ \inf b(t,P)>b_0>0, \forall (t,P)\in [0,T]\times S, &{} \hbox {} \\ \exists \eta \in (\frac{1}{2}, 1), |b(t,P)-b(s,P)|\le c |t-s|^\eta , \forall P\in S, &{} \hbox {} \end{array} \right. \end{aligned}$$

and $E_S$ is the energy functional on S introduced in (3.5). T is a fixed positive real number, $\phi$ is a given function in $L^2(Q)$. We assume that for every $t\in [0,T],$ J is a mapping from $L^{2p}(Q)\rightarrow L^2(Q), p>1,$ locally Lipschitz , i.e., Lipschitz on bounded sets in $L^{2p}(Q)$ see condition 5.3. For all $t\in [0,T], A(t)$ is the infinitesimal generator of the analytic contraction (positivity preserving) semigroup $T_t(s)$ from $L^2(Q)$ into $L^2(Q),$ associated to E[t, u], (see Proposition 3.6). Moreover, there exists a unique evolution operator governing A, $U(t,s):\Diamond \rightarrow \mathcal {L}(L^2(Q))$ where $\Diamond =\{((t,s): 0\le s \le t\le T\}$.

We study problem (P) looking for the mild solution

$$\begin{aligned} u(t)= U(t,0)\phi +\int _0^t U(t,s) J(u(s)) ds. \end{aligned}$$

(1.3)

In order to prove the existence of the solutions to (1.3), the usual way is to use a contraction argument in suitable Banach spaces see, e.g., [23]. In the autonomous case, when dealing with a semigroup approach in regular domains, the functional setting is that of an interpolation space between the domain of the generator A and $L^2(Q)$ or the domain of a fractional power of A, we refer the reader to [6, 16, 23, 27, 29], as well as in the nonautonomous case [2]. In our fractal case, we do not know the domain of A,  even in the autonomous case. We stress the fact that it is not known a characterization of the domain of the fractal Laplacian $\Delta _S.$ Here, we extend to the nonautonomous case the approach developed in [21] for the autonomous case. We adapt the abstract approach in [33] to deal with the semilinearity and to prove local existence and uniqueness results for the mild solution. The key tool in [33] is an assumption on the estimate of the evolution operator U(t, s) as a bounded operator from $L^2(Q)$ to $L^{2p}(Q)$ which we prove using the techniques in [10].

Energy functionals of type (1.2) appear when considering the variational approach to Wentzell problems in fractal domains known also as problems with dynamical boundary conditions [31]. Wentzell problems appear in different frameworks, for instance, in electrostatics or magnetostatics, the model problem which describes the heat transfer through an infinitely conductive layer (in this regard see the [28] and the references listed there). Also in the different field of “hydraulic fracturing”—a technique used in order to increase the flow of oil from a reservoir into a producing oil well—the model problem is a transmission problem (see [4]). For other applications, see [22].

The literature on boundary value problems with dynamical conditions is huge; we refer to [15] for a derivation of such boundary conditions and to [12] and the references listed in.

All these papers deal with smooth domains. The case of irregular domains is studied in [9, 21, 22, 32] and more recently in [7, 8] for fractional operators.

The layout of this paper is the following. In Sect. 2, we recall the preliminaries on the geometry and the functional spaces. In Sect. 3, we consider the energy forms and the associated evolution operators. In Proposition 3.7 and in Theorem 3.8, we prove crucial properties of the generators and the evolution operators associated with the form. In Sect. 4, we prove the ultracontractivity of the evolution operator in Theorem 4.9 which relies on Theorem 4.1, 4.4 and Nash inequality. In Sect. 5, we consider the abstract Cauchy problem (P) and we prove local existence results in Theorem 5.2. In Theorem 5.4, we prove regularity results for the mild solution. Finally, in Theorem 5.5 we prove global existence results under suitable assumptions on the initial datum.

2 Preliminaries

2.1 Geometry

In the paper, we denote by $P=(x_1,x_2,x_3)$ points in $\mathbb {R}^3$, by $|P-P_0|$ the Euclidean distance and by $B(P_0,r)=\{P\in \mathbb {R}^3: |P-P_0|<r \},\, P_0\in \mathbb {R}^3, r>0,$ the Euclidean balls. By the Koch snowflake F, we will denote the union of three coplanar Koch curves $K_i$ (see [11]).

The Hausdorff dimension of the Koch snowflake is given by $d_f=\frac{\ln 4}{\ln 3}$. This fractal is no longer self-similar (and hence, not nested). One can define, in a natural way, a finite Borel measure $\mu$ supported on $\partial \Omega$ by

$$\begin{aligned} \mu :=\mu _1+\mu _2+\mu _3, \end{aligned}$$

(2.1)

where $\mu _i$ denotes the normalized $d_f$-dimensional Hausdorff measure, restricted to $K_i$, $i=1,2,3$.

Now we introduce the notion of $d-set$ according to [17]:

Definition 2.1

Let $\mathcal {S}\subset \mathbb {R}^n$ be a closed non-empty subset. It is a d-set $(0<d\le n)$ if there exists a Borel measure ${\tilde{\mu }}$ with $supp {\tilde{\mu }}=\mathcal {S}$ such that for some constants $c_1=c_1(\mathcal {S})>0$ and $c_2=c_2(\mathcal {S})>0$

$$\begin{aligned} c_1r^{d}\le {\tilde{\mu }}(B(P,r))\le c_2r^{d}\ \;(P\in \mathcal {S}, 0<r\le 1). \end{aligned}$$

(2.2)

Such a ${\tilde{\mu }}$ is called a d-measure on $\mathcal {S}$.

Proposition 2.2

The set F is a d-set with $d=d_f$. The measure $\mu$ is a d-measure.

See [13, 25].

We denote by $\Omega$ the (open) two-dimensional snowflake domain.

Fig. 1

The snowflake domain $\Omega$

Let Q denote the open bounded set defined as the Cartesian product of the snowflake domain $\Omega$ and the unit interval; the “lateral surface” S is the product of the Koch snowflake F and $I=[0,1]$, and the bases are the sets $\Omega \times \{0\}$ and $\Omega \times \{1\}$ (see Fig. 2). For details on the fractal sets, see [21].

Fig. 2

The fractal domain Q

We give a point $P\in S$ the Cartesian coordinates $P=(x,y),$ where $x=(x_1,x_2)$ are the coordinates of the orthogonal projection of P on the plane containing F and y is the coordinate of the orthogonal projection of P on the $y-$line containing the interval I: $P=(x,y)\in S, x=(x_1,x_2)\in F, y\in I.$

We introduce on S the measure

$$\begin{aligned} \textrm{d}m=\textrm{d}\mu \times \textrm{d}y, \end{aligned}$$

(2.3)

where $\mu$ is the $d_f$-normalized Hausdorff measure on $\partial \Omega$ (see [11, 13]), $d_f:=\frac{\ln 4}{\ln 3}$ is the Hausdorff dimension of $\partial \Omega$ and $\textrm{d}y$ is the one-dimensional Lebesgue measure on I. We remark that S is a $(d_f+1)$-set, while the boundary $\partial Q=S\cup (\Omega \times \{0\})\cup (\Omega \times \{1\})$ is neither a 2-set nor a $(d_f+1)$-set; $\partial Q$ is a closed set of $\mathbb {R}^3$.

2.2 Functional spaces

By $L^p(\cdot ), p>1$, we denote the Lebesgue space with respect to the Lebesgue measure on subsets of $\mathbb {R}^3$, which will be left to the context whenever that does not create ambiguity. Let $\mathcal {T}$ be a closed set of $\mathbb {R}^3$, by $C(\mathcal {T})$ we denote the space of continuous functions on $\mathcal {T}$, and by $C_0(\mathcal {T})$ we denote the space of continuous functions vanishing on $\partial \mathcal {T}$. Let $\mathcal {G}$ be an open set of $\mathbb {R}^3$, by $H^1(\mathcal {G})$ we denote the usual Sobolev spaces (see Necǎs [26]); by $H^s(\mathcal {G})$, where $s\in \mathbb {R}^+$ we denote the usual (possibly fractional) Sobolev spaces (see [26]); $H^s_0(\mathcal {G})$ is the closure of $\mathcal {D}(\mathcal {G})$, (the infinitely differentiable functions with compact support on $\mathcal {G}$), with respect to the $\Vert \cdot \Vert _{H^s}$-norm.

We now recall a trace theorem. For f in $H^s(\mathcal {G})$, we put

$$\begin{aligned} \gamma _0f(P)=\lim _{r\rightarrow 0}{1\over |B(P,r)\cap \mathcal {G}^|}\int _{B(P,r)\cap \mathcal {G}}f(Q)\textrm{d}\mathcal {L}_2 \end{aligned}$$

(2.4)

at every point $P\in \overline{\mathcal {G}}$ where the limit exists. It is known that the limit (2.4) exists at quasi every $P\in \overline{\mathcal {G}}$ with respect to the (s, 2)-capacity [1].

We now come to the definition of the Besov spaces. Actually, there are many equivalent definitions of these spaces; see, for instance, [17, 30]. We recall here the one which best fits our aims and we will restrict ourselves to the case $0<\alpha <1$, $p=q=2$; the general setting is much more involved, see [17]. By $B^{2,2}_\alpha (S)$, we denote the space of functions

$$\begin{aligned} B^{2,2}_\alpha (S)=\{ u\in L^2(S): \Vert u\Vert _{B^{2,2}_\alpha (S)}< +\infty \} \end{aligned}$$

where

$$\begin{aligned} \Vert u\Vert _{B^{2,2}_\alpha (S)}=\Vert u\Vert _{L^2(S)}+ \Big ( \int \int _{|x-y|\le 1} \frac{|u(x)-u(y)|^2}{|x-y|^{2\alpha +d_f+1}}\textrm{d}m(x) \textrm{d}m(y) \Big )^{1\over 2} \end{aligned}$$

Theorem 2.3

Let $\alpha ={d_f\over 2}.$ Then, $B^{2,2}_{\alpha }({S})$ is the trace space of $H^1(Q)$ in the following sense:

• (i) $\gamma _0$ is a continuous linear operator from $H^{1}(Q)$ to $B^{2,2}_{\alpha }(S)$,

• (ii) there is a continuous linear operator $\textrm{Ext}$ from $B^{2,2}_{\alpha }({S})$ to $H^{1}(Q)$ such that $\gamma _0\circ \textrm{Ext}$ is the identity operator in $B^{2,2}_{\alpha }({S}).$

For the proof, we refer to Theorem 1 of Chapter V in [17], see also [30].

Throughout the paper, c will denote different constants.

3 Energy forms and associated evolution operators

In this section, we introduce the energy functional on S. We first define the energy functional on the cross section F by integrating its Lagrangian on F. For the concept of Lagrangian on fractals, i.e., the notion of a measure-valued local energy, we refer to [14, 24]. Here for the sake of simplicity, we only mention that the Lagrangian on F, $\mathcal {L}_F$ is a measure valued map on $\mathcal {D}(F)\times \mathcal {D}(F)$ which is bilinear symmetric and positive ($\mathcal {L}_F[u]$ is a positive measure.) The measure-valued Lagrangian takes on the fractal F the role of the Euclidean Lagrangian $d\mathcal {L}(u,v)= \textrm{D} u \cdot \textrm{D} v dx.$ Note that in the case of the Koch curve, the Lagrangian is absolutely continuous with respect to the measure $\mu .$

On the contrary, this is not true on most fractals (see [24]). In [13], the Lagrangian $\mathcal {L}_F$ on the snowflake F has been defined by using its representation as a fractal manifold. Here, we do not give details on the construction and definition of $\mathcal {L}_F$ and we refer to Sect. 4 in [13] for details; in particular, in Definition 4.5, a Lagrangian measure $\mathcal {L}_F$ on F and the corresponding energy form $\mathcal {E}_F$ as

$$\begin{aligned} \mathcal {E}_F(u,v)=\int _F\textrm{d}\mathcal {L}_F(u,v) \end{aligned}$$

(3.1)

with domain $\mathcal {D}(F)$ have been introduced. The domain $\mathcal {D}(F)$, which is a Hilbert space with norm

$$\begin{aligned} (\Vert u\Vert _{L^2(F,\mu _F)}^2+\mathcal {E}_F(u,u))^\frac{1}{2} \end{aligned}$$

has been characterized in terms of the domains of the energy forms on $K_i$ (see [13] Theorem 4.6). In the following, we will omit the subscript F, the Lagrangian measure will be simply denoted by $\mathcal {L}(u,v)$, and we will set $\mathcal {L}[u]=\mathcal {L}(u,u)$; an analogous notation will be adopted for the energies. We define the energy forms $E_S$ on the fractal layer $S=F\times I$ by setting

$$\begin{aligned} E_S[u]=\sigma ^1\int _I\int _F\mathcal {L}_x[u](\textrm{d}x)\textrm{dy}+\sigma ^2\int _F\int _I|D_yu|^2\textrm{d}y\mu _F(\textrm{d}x) \end{aligned}$$

(3.2)

where $\sigma ^1$ and $\sigma ^2$ are positive constants. Here, $\mathcal {L}_x(\cdot ,\cdot )(\textrm{d}x)$ denotes the measure–valued Lagrangian (of the energy form $\mathcal {E}_F$ of F with domain $\mathcal {D}(F)$) now acting on u(x, y) and v(x, y) as function of $x\in F$ for a.e. $y\in I$; $\mu _F(\textrm{d}x)$ is the Hausdorff measure acting on each section F of S for a.e. $y\in I$ with $d_f=\frac{\log 4}{\log 3}$, $D_y(\cdot )$ denotes the derivative in the y direction.

The form $E_S$ is defined for $u\in \mathcal {D}(S)$ where $\mathcal {D}(S)$ is the closure in the intrinsic norm

$$\begin{aligned} \Vert u\Vert _{\mathcal {D}(S)}=(E_S[u]+\Vert u\Vert ^2_{L^2(S,m)})^{\frac{1}{2}} \end{aligned}$$

(3.3)

of the set

$$\begin{aligned} C_0(S)\cap L^2(0,1;\mathcal {D}(F))\cap H_0^1(0,1;L^2(F)) \end{aligned}$$

(3.4)

where $L^2(F)=L^2(F,\mu _F(\textrm{d}x))$.

In the following, we shall also use the form $E_S(u,v)$ which is obtained from $E_S[u]$ by the polarization identity:

$$\begin{aligned} E_S(u,v)=\frac{1}{2}\{E_S[u+v]-E_S[u]-E_S[v]\},\ u,v\in \mathcal {D}(S). \end{aligned}$$

(3.5)

It can be proved as in Proposition 3.1 of [25]:

Proposition 3.1

In the previous notations and assumptions, the form $E_S$ with domain $\mathcal {D}(S)$ is a regular Dirichlet form in $L^2(S,m)$ and the space $\mathcal {D}(S)$ is a Hilbert space under the intrinsic norm (3.3).

For the definition and properties of regular Dirichlet forms, we refer to [14]. We now define the Laplace operator on S. As $(E_S,\mathcal {D}(S))$ is a closed, bilinear form on $L^2(S,m)$, there exists (see Chap. 6, Theorem 2.1 in [18]) a unique self-adjoint, non-positive operator $\Delta _S$ on $L^2(S,m)$, with domain $\mathcal {D}(\Delta _S)\subseteq \mathcal {D}(S)$ dense in $L^2(S,m)$, such that

$$\begin{aligned} E_S(u,v)=-\int _S(\Delta _Su)v\textrm{d}m,\ u\in \mathcal {D}(\Delta _S), v\in \mathcal {D}(S). \end{aligned}$$

(3.6)

Let $(\mathcal {D}(S))'$ denote the dual of the space $\mathcal {D}(S)$. We now introduce the Laplace operator on the fractal S as a variational operator from $\mathcal {D}(S)\rightarrow (\mathcal {D}(S))'$ by

$$\begin{aligned} E_S(z,w)=-\langle \Delta _Sz,w\rangle _{(\mathcal {D}(S))',\mathcal {D}(S)} \end{aligned}$$

(3.7)

for $z\in \mathcal {D}(S)$ and for all $w\in \mathcal {D}(S),$ where $\langle \cdot ,\cdot \rangle _{(\mathcal {D}(S))',\mathcal {D}(S)}$ is the duality pairing between $(\mathcal {D}(S))'$ and $\mathcal {D}(S)$. We use the same symbol $\Delta _S$ to define the Laplace operator both as a self-adjoint operator in (3.6) and as a variational operator in (3.7). It will be clear from the context to which case we refer.

Consider now the space of functions $u:Q\rightarrow \mathbb {R}$

$$\begin{aligned} V(Q,S)=\left\{ u\in H^1(Q)\,\ u|_S\in \mathcal {D}(S), u=0 \;\; \text{ on }\;\;\;\; \tilde{\Omega }\right\} . \end{aligned}$$

(3.8)

V(Q, S) is nontrivial see proposition 3.3 of [20]. We now introduce the energy form $E_0[u]:V(Q,S)\rightarrow \mathbb {R}$:

$$\begin{aligned} E_0[u]=\int _Q|\textrm{D}u|^2\textrm{d}Q+ E_S[u|_S] \end{aligned}$$

(3.9)

defined on the domain V(Q, S). In the following, by $E_0(u,v)$ we will denote the corresponding bilinear form

$$\begin{aligned} E_0(u,v)=\int _Q\textrm{D}u\textrm{D}v\textrm{d}Q+{E_S(u|_S,v|_S)} \end{aligned}$$

(3.10)

defined on $V(Q,S)\times V(Q,S)$. We now introduce the energy form

$$\begin{aligned} E[t,u]=E_0[u]+ \int _S b(t,P) |u|_S|^2 \, \textrm{d}m \end{aligned}$$

(3.11)

defined on the domain $[0,T]\times V(Q,S)$.

By E(t, u, v), we will denote the corresponding bilinear form

$$\begin{aligned} E(t,u,v)=E_0(t,u,v)+ \int _S b(t,P) u|_S v|_S \, \textrm{d}m \end{aligned}$$

(3.12)

defined on $[0,T]\times V(Q,S)\times V(Q,S)$. Here, $b(t,P):(0,T)\times S\rightarrow \mathbb {R}$ is a function such that

$$\begin{aligned} A) \left\{ \begin{array}{ll} b\in L^\infty ([0,T]\times S), &{} \hbox {} \\ \inf b(t,P)>b_0>0, \forall (t,P)\in [0,T]\times S, &{} \hbox {} \\ \exists \eta \in (\frac{1}{2}, 1): |b(t,P)-b(s,P)|\le c |t-s|^\eta , \forall P\in S. &{} \hbox {} \end{array} \right. \end{aligned}$$

Proposition 3.2

For every $t\in [0,T]$, the form E defined in (3.12) is closed in $L^2(Q)$ and the space V(Q, S) is a Hilbert space equipped with the scalar product

$$\begin{aligned} (u,v)_{V(Q,S)}= (\textrm{D}u,\textrm{D}v)_{L^2(Q)}+ E_S(u,v)+(u,v)_{L^{2}(S)} \end{aligned}$$

(3.13)

We denote by $\Vert u\Vert _{V(Q,S)}$ the norm in V(Q, S),  associated with (3.13), that is

$$\begin{aligned} \Vert u\Vert _{V(Q,S)}=(\Vert \textrm{D}u\Vert _{L^2(Q)}^2+\Vert u\Vert _{\mathcal {D}(S)}^2)^{1/2}. \end{aligned}$$

(3.14)

As in Proposition (3.6) and (3.1) in [20], the following can be proved.

Proposition 3.3

The space $\mathcal {D}(S)$ is embedded in $B_{\beta ,0}^{2,2}(S)$, $\beta =\frac{d_f}{2}.$

Proposition 3.4

The space $\mathcal {D}(S)$ is embedded in $B_{\alpha }^{2,2}(S)$, $\alpha < 1.$

For every $t\in [0,T],$ (E, V(Q, S)) is a closed bilinear form on $L^2(Q)$, with domain V(Q, S) dense in $L^2(Q)$, there exists (see Chapter 6 Theorem 2.1 in [18]) a unique self-adjoint non-positive operator A(t) on $L^2(Q)$, with domain $\mathcal {D}(A(t))\subseteq V(Q,S)$ dense in $L^2(Q)$, such that

$$\begin{aligned} E(t,u,v) = -\int _Q A(t)\;u\; v \, \textrm{d}Q,\;u\in \mathcal {D}(A(t)), v\in V(Q,S). \end{aligned}$$

(3.15)

Proposition 3.5

For every $t\in [0,T]$, the form E(t, u, v) is bounded and coercive in V(Q, S); there exist c, $\alpha \in \mathbb {R}^+$ such that:

$$\begin{aligned}{} & {} E(t,u,v)\le c\Vert u\Vert _{V(Q,S) }\Vert v\Vert _{V(Q,S)}, \\{} & {} E[t,u]\ge \alpha \Vert u\Vert ^2_{V(Q,S)}, \end{aligned}$$

and it has the square root property, that is, ${\mathcal {D}}(A(t))^{\frac{1}{2}}=V(Q,S).$ Moreover, there exists $c>0$ such that for every $\eta \in (1/2,1),$

$$\begin{aligned} |E(t,u,v)-E(s,u,v)|\le c |t-s|^\eta \Vert u\Vert _{V(Q,S)} \Vert v\Vert _{V(Q,S)}, \, 0\le s,t\le T. \end{aligned}$$

(3.16)

Proof

$E[t,u]\ge E_0[u]+b_0\Vert u\Vert ^2_{L^2(S)}\ge \min \{1,b_0\}\Vert u\Vert ^2_{V(Q,S)}.$

Since the form is symmetric and bounded, the square root property follows. As to (3.16), we note that

$$\begin{aligned}{} & {} |E(t,u,v)-E(s,u,v)|\le \int _{S} |(b(t,\cdot )-b(s,\cdot )) u(\cdot ) v(\cdot ) dm |\\{} & {} \quad \le c |t-s|^\eta \Vert u\Vert _{V(Q,S)} \Vert v\Vert _{V(Q,S)}; \end{aligned}$$

the last inequality follows from the hypothesis A) on b. $\square$

Proposition 3.6

For every $t \in [0,T]$ and $\tau \ge 0$, $A(t):\mathcal {D}(A(t))\rightarrow L^2(Q)$ is the generator of a semigroup $e^{\tau A(t)}$ on $L^2(Q)$ which is strongly continuous, contractive and analytic with angle $\omega _{A(t)}>0$.

Proof

The contraction property follows from Lumer–Phillips Theorem and analyticity follows from the coercivity. $\square$

Proposition 3.7

The operator A(t) for every $t\in [0,T]$ satisfies the following properties.

• 1. The spectrum of A(t) is contained in a sectorial open domain

  $$\begin{aligned} \sigma (A(t))\subset \Sigma _\omega =\{ \lambda \in \mathbb {C}, |arg \lambda |<\omega \} \end{aligned}$$

  for some fixed angle $0<\omega <\pi /2.$ The resolvent satisfies the estimate:

  $$\begin{aligned} \Vert (\lambda -(A(t))^{-1}\Vert _{\mathcal {L}(L^2(Q))}\le M/ |\lambda | \end{aligned}$$

  for $M\ge 1$ independent from t and $\lambda \notin \Sigma _\omega \cup 0;$ moreover A(t) is invertible and $\Vert (A(t))^{-1}\Vert \le M_1$ with $M_1$ independent of t.

• 2. $\mathcal {D}(A(t))\subset \mathcal {D}(A(s))^{\frac{1}{2}}=V(Q,S)), 0\le s\le t\le T,$ in particular $\mathcal {D}(A(t))\subset \mathcal {D}(A(s))^{\nu }, \forall \nu : 0<\nu \le \frac{1}{2}.$

• 3. $(A(t))^{-1}$ is Hölder continuous in t, in the sense of Yagi,

  $$\begin{aligned} \Vert (A(t))^{\frac{1}{2}} \Big ( A(t)^{-1}-A(s)^{-1}\Big )\Vert _{\mathcal {L}(L^2(Q))}\le c |t-s|^\eta \end{aligned}$$

  (3.17)

  with some fixed exponent $\eta \in (1/2,1],$ and $c>0.$

Proof

The first two properties follow from Propositions 3.5 and 3.6. To prove the Hölder continuity, one can proceed as in [35] (Chapter 3, section 7.1 ).

Let $\mathcal {A}(t):V(Q,S)\rightarrow V(Q,S)'$ denote the sectorial operator with angle $\omega _{\mathcal {A}(t)}\le \omega _A <\pi /2$ associated with E(t, u, v):

$$\begin{aligned} E(t,u,v)=-<\mathcal {A}(t)u, v>_{V(Q,S)',V(Q,S)} \end{aligned}$$

with $u\in \mathcal {D}(\mathcal {A}(t))=V(Q,S)$ Let $\phi \in V(Q,S)'$ and $u\in V(Q,S)$. We have

$$\begin{aligned}{} & {} <\mathcal {A}(t)[ \mathcal {A}(t)^{-1}- \mathcal {A}(s)^{-1}]\phi ,u>_{V(Q,S)',V(Q,S)} \\{} & {} =-< [\mathcal {A}(t)-\mathcal {A}(s)]\mathcal {A}(s)^{-1}\phi ,u>=+ E(t,\mathcal {A}(s)^{-1}\phi , u)- E(s,\mathcal {A}(s)^{-1}\phi , u), \end{aligned}$$

from (3.16) we obtain

$$\begin{aligned} \Vert \mathcal {A}(t)[\mathcal {A}(t)^{-1}- \mathcal {A}(s)^{-1}]\phi \Vert _{L^2(Q)}\le c |t-s|^\eta \Vert \mathcal {A}(s)^{-1}\Vert _{\mathcal {L}(V(Q,S)\rightarrow V(Q,S)')}\Vert \phi \Vert _{V(Q,S)'}, \end{aligned}$$

from [19] page 190, we have that $\Vert \mathcal {A}(s)^{-1}\Vert _{\mathcal {L}(V(Q,S)\rightarrow V(Q,S)')} \le c.$ We conclude that

$$\begin{aligned} \Vert \mathcal {A}(t)(\mathcal {A}(t)^{-1}- \mathcal {A}(s)^{-1})\phi \Vert _{V(Q,S)'}\le c |t-s|^\eta \Vert \phi \Vert _{V(Q,S)'}. \end{aligned}$$

In order to prove condition (3.17), we note that for every $z\in V(Q,S),$ $A^{\frac{1}{2}}(\cdot )z= \mathcal {A}^{-1/2}(\cdot )\mathcal {A}(\cdot )z,$

$$\begin{aligned}{} & {} (A(t)^{\frac{1}{2}}[A(t)^{-1}-A(s)^{-1}]\phi , u)\\{} & {} \quad = \Big ( E(t,A(s)^{-1}\phi , ({\mathcal A}(t)^{-\frac{1}{2}})' u)- E(s,A(s)^{-1}\phi , ({\mathcal A}(t)^{-\frac{1}{2}})'u) \Big ) \end{aligned}$$

Since the adjoint operators have the same norm, taking into account [19] pag 190, condition (3.17) holds. $\square$

From the above results we deduce

Theorem 3.8

For every $t\in [0,T]$, let $A(t):{\mathcal {D}}(A(t))\rightarrow L^2(Q)$ be the linear unbounded operator defined in (3.15). Then, there exists a unique family of evolution operators $U(t,s)\in \mathcal {L}(L^2(Q))$ such that

• 1. $U(s,s)=I,\;\; 0\le s\le T,$

• 2. $U(t,s)U(s,\sigma )=U(t,\sigma ), 0\le \sigma \le s\le t\le T,$

• 3.
  $$\begin{aligned} \Vert U(t,s)\Vert _{\mathcal {L}(L^2(Q))}\le 1, 0\le s\le t\le T, \end{aligned}$$

  (3.18)

  and U(t, s) for $0\le s \le t <T$ is a strongly contractive family on $L^2(Q).$

• 4. $t\rightarrow U(t,s)$ is differentiable in (s, T] with values in $\mathcal {L}(L^2(Q))$ and $\frac{\partial U(t,s)}{\partial t}= A(t)U(t,s).$

• 5. A(t)U(t, s) is an $\mathcal {L}(L^2(Q))-$valued continuous function for $0\le s<t\le T.$ Moreover, there exists $c>0$ such that

  $$\begin{aligned} \Vert A(t)U(t,s)\Vert _{\mathcal {L}(L^2(Q))}\le \frac{c}{(t-s)}, \; 0\le s< t <T. \end{aligned}$$

  (3.19)

For the proof, see section 5.3 in [35].

Consider now the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} u_t=A(t)u(t), &{} \hbox {for}\;\, 0<t\le T \\ u(0)=u_0, &{} \hbox {} \end{array} \right. \end{aligned}$$

(3.20)

with $u_0\in L^2(Q).$

Theorem 3.9

For every $u_0\in L^2(Q)$, there exists a unique $u\in C([0,T]; L^2(Q))\cap C^1((0,T]; L^2(Q)),$ $A(t)u\in C((0,T]; L^2(Q))$ such that

$$\begin{aligned} \Vert u(t)\Vert _{L^2(Q))} + t\Vert \frac{d u(t)}{dt}\Vert _{L^2(Q))} + t \Vert A(t)u(t)\Vert _{L^2(Q))}\le c \Vert u_0\Vert _{L^2(Q))}, \end{aligned}$$

for $0<t\le T,$ where c is a positive constant.

Moreover,

$$\begin{aligned} u(t)=U(t,0)u_0 \end{aligned}$$

For the proof, see Theorem 3.9 in [35]).

4 Ultracontractivity property

Following Theorems 5.1 and 5.2 in Daners [10] and taking into account the contraction property of U(t, s), we prove ultracontractivity properties for $p\ge 1.$ We start by analyzing the case $p\ge 2.$

Theorem 4.1

For every $p\in [2,+\infty ]$, there exists an operator $U_p(t,s)\in \mathcal {L}(L^p(Q))$ such that

$$\begin{aligned} U_p(t,s)u_0=U(t,s) u_0, u_0\in L^p(Q) \end{aligned}$$

Moreover, for every $s\ge 0$ the map $U_p(\cdot ,s)$ is strongly continuous from $(s, \infty )$ to $\mathcal {L}(L^p(Q)), \forall t\ge s.$

$$\begin{aligned} \Vert U_p(t,s)\Vert _{\mathcal {L}(L^p(Q))}\le 1, \; \; p\ge 2 \end{aligned}$$

(4.1)

Proof

We start proving that if $u_0\in L^p(Q),$ for some $p> 2;$ then there exists an operator $U_p(t,s)\in \mathcal {L}(L^p(Q))$ such that $U_p(t,s)u_0= U(t,s)u_0$ and $\Vert U(t,s)u_0\Vert _{L^p(Q)}\le \Vert u_0\Vert _{L^p(Q)}$.

We set $u(t)=U(t,0)u_0$ and

$$\begin{aligned}{} & {} w_{k,p}(u)= \left\{ \begin{array}{ll} |u|^{\frac{p}{2}}, &{} \hbox {if }|u|\le k \\ (k^{\frac{p}{2}}+ p/2 k^{\frac{p}{2}-1}(|u|-k)), &{} \; \hbox {if};|u|> k \end{array} \right. \nonumber \\{} & {} \quad v_{k,p}(u)= \left\{ \begin{array}{ll} p/2 \; \text{ sign } u \;|u|^{{p-1}}, &{} \hbox {if }|u|\le k \\ p/2 \; \text{ sign }u (k^{p-1}+ p/2 k^{p-2}(|u|-k)), &{} \; \hbox {if}\;|u|> k \end{array} \right. \end{aligned}$$

(4.2)

$v_{k,p}(u)=w_{k,p}(u)w_{k,p}'(u),$ $w_{k,p}(u),v_{k,p}(u) \in C^0(\mathbb {R})$, $w_{k,p}'(u),v_{k,p}'(u)\in L^\infty (\mathbb {R})$ and are continuous for $|u|\ne \pm k.$

From Lemma 3.3 in [10], we have

$$\begin{aligned} 1/2 (\Vert w_{k,p}(u(t))\Vert ^2_{L^2(Q)}-\Vert w_{k,p}(u(0))\Vert ^2_{L^2(Q)})= \int _{0}^{t}(u_t(s),v_{k,p}(u(s)))_{L^2(Q)}ds \end{aligned}$$

and since u is a weak solution of the Cauchy problem

$$\begin{aligned} 1/2 (\Vert w_{k,p}(u(t))\Vert ^2_{L^2(Q)}-\Vert w_{k,p}(u(0))\Vert ^2_{L^2(Q)}) =-\int _{0}^{t} E(s,u(s),v_{k,p}(u(s))) ds \end{aligned}$$

for every $k\ge 1, p\ge 2$. From the absolute continuity of $w_{k,p}(u)$ after differentiation

$$\begin{aligned}{} & {} \frac{d \Vert w_{k,p}(u(t))\Vert ^2_{L^2(Q)} }{dt}= -2E(t,u(t),v_{k,p}(u(t))) \nonumber \\{} & {} \quad =-2E_0(t,u(t),v_{k,p}(u(t))- 2\int _{S}b(t) u(t)v_{k,p}(u(t))dm. \end{aligned}$$

(4.3)

Since $uv_{k,p}(u)\ge 0$, $b(t)>0$ we have $-\int _{S}b(t) u(t)v_{k,p}(u(t))dm\le 0$; moreover, $E(s,u(s),v_{k,p}(u(s))=v_{k,p}'(u(s)) E(s,u(s),u(s))$ and as $v_{k,p}'(u(s))>0, \forall u\ne \pm k$ we have that

$$\begin{aligned} \frac{d \Vert w_{k,p}(u(t))\Vert ^2_{L^2(Q)} }{dt}\le 0 \end{aligned}$$

hence $\Vert w_{k,p}(u(t))\Vert ^2_{L^2(Q)}$ is decreasing in t, thus if $t>0$ $\Vert w_{k,p}(u(t))\Vert ^2_{L^2(Q)}\le \Vert w_{k,p}(u(0))\Vert ^2_{L^2(Q)}=\Vert w_{k,p}(u_0)\Vert ^2_{L^2(Q)}.$ As $w_{k,p}(u)\rightarrow |u|^{\frac{p}{2}}, k\rightarrow +\infty$ increasing, from monotone convergence Theorem we have $\Vert |u|^{\frac{p}{2}}\Vert ^2_{L^2(Q)} \le \Vert u_0^{\frac{p}{2}}\Vert ^2_{L^2(Q)}$ hence $\Vert U(t,0)u_0\Vert _{L^p(Q)}\le \Vert u_0\Vert _{L^p(Q)}.$ Therefore, it is possible to regard U(t, 0) from $L^p$ to $L^p, \, \forall p\ge 2;$ as a contraction operator. It clearly satisfies (4.1) for $s=0$; the inequality holds also for $s\ne 0.$ To prove the continuity, it is sufficient to prove it for $p=2;$ the other cases can be deduced from (4.1) and Riesz–Thorin Interpolation Theorem, the proof follows line by line theorem 5.1 in [10]. $\square$

We now prove that U(t, s) maps $L^2(Q)$ into $L^\infty (Q)$.

The proof relies on Nash inequality which in turn is based on Sobolev inequality. Since Q is a John domain from [3] (see also [5]) we have

Proposition 4.2

For every $u\in W^{1,p}(Q), 1<p<n,$

$$\begin{aligned} \Vert u\Vert _{L^{p^*}(Q)}\le c \Vert u\Vert _{W^{1,p}(Q)} \end{aligned}$$

where c depends on p, n, Q and $p^*=\frac{np}{n-p}$.

By proceeding as in Proposition 4.5 in [10] and taking into account that $u\in V(Q,S)$ from the Poincarè inequality, we can prove

Proposition 4.3

Let u be in V(Q, S). Then, the Nash inequality holds:

$$\begin{aligned} \Vert u\Vert _{L^2(Q)}^{2+ \frac{4}{n}}\le C_N \Vert \textrm{D}u\Vert ^2_{L^2(Q)}\Vert u\Vert _{L^1(Q)}^{\frac{4}{n}}. \end{aligned}$$

(4.4)

Finally, we have:

Theorem 4.4

Let $p\ge 2.$ The evolution operator $U_p(t,s)$ satisfies the following property. There exists a $c\in \mathbb {R}^+$ depending on n such that

$$\begin{aligned} \Vert U_p(t,s)u(s)\Vert _{L^\infty (Q)}\le c (t-s)^{-\frac{n}{4}}\Vert u(s)\Vert _{L^2(Q)} \end{aligned}$$

for every $u\in L^2(Q),$ and $0\le s<t<T$

In order to prove this theorem, we will prove some preliminary results.

Since $u\in L^2(0,T,V(Q,S)),$ $u_t\in L^2(0,T,(V(Q,S))'),$ then $w_{k,p}(u),v_{k,p}(u)\in L^2(0,T,V(Q,S)), w_{k,p}'(u), v_{k,p}'(u)\in L^2(0,T,(V(Q,S))')$.

By differentiating

$$\begin{aligned}{} & {} \Vert \textrm{D} w_{k,p}(u)\Vert ^2_{L^2(Q) }=\int _{Q }|\textrm{D} w_{k,p}(u)|^2 dQ = \int _{Q } p/(2(p-1))\textrm{D}u \textrm{D}v_{k,p} dQ\le \\{} & {} p/(2(p-1)) E_0(u,v_{k,p})< E(u,v_{k,p}) < 2 E(u,v_{k,p}) = -\frac{d \Vert w_{k,p}\Vert ^2_{{L^2(Q)}}}{dt} \end{aligned}$$

Hence,

$$\begin{aligned} \frac{d \Vert w_{k,p}\Vert ^2_{{L^2(Q)}}}{dt}\le - \Vert \textrm{D} w_{k,p}(u)\Vert ^2_{L^2(Q) }. \end{aligned}$$

(4.5)

We now fix $k\ge 1$ and set for arbitrary $p\ge 2,$

$$\begin{aligned} \phi _p(t):=\Vert w_{k,p}(t)\Vert _{L^2(Q) }^{\frac{2}{p}} \end{aligned}$$

where $w_{k,p}(t)$ (see (4.2)) and $u(\cdot )$ the solution of (3.20) with $u_0\ne 0\in L^2(Q)$. As $\phi _p$ is locally absolutely continuous and thus differentiable a.e.

As (3.20) has a unique solution it follows that $u(t)=0$ for all $t\ge t_0$ if $u(t_0)=0$ for some $t_0>0$. Hence, there exists an interval $J:=[0,t_0)$ such that $\phi _p(t)\ne 0$ for all $t\in [0,t_0)$ and $\phi _p(t)= 0$ for $t\ge t_0.$

We now state a fundamental lemma.

Lemma 4.5

For all $p\ge 2$ and all $t\in J$, there exists $c\in \ R$ such that

$$\begin{aligned} (\phi _{2p}(t))^{-\frac{4p}{n}}\ge \frac{2 c}{n}\int _{0}^{t} \Big (\phi _p(s)\Big )^{-\frac{4p}{n}} ds. \end{aligned}$$

Proof

From (4.5), we have that

$$\begin{aligned} \frac{d \Vert w_{k,2p}\Vert ^2_{{L^2(Q)}}}{dt}\le - \Vert \textrm{D} w_{k,2p}(u)\Vert ^2_{L^2(Q) }\le - c (\Vert w_{k,2p}\Vert _{L^1(Q)})^{-\frac{4}{n}} (\Vert w_{k,2p}\Vert _{L^2(Q)})^{2+\frac{4}{n}} \end{aligned}$$

where the last inequality follows from (4.4). Since $(\Vert w_{k,2p}\Vert ^2_{{L^1(Q)}})^{\frac{1}{p}}\le \phi _{p}(t)$ and $\phi _{2p}(t)=\Vert w_{k,2p}\Vert _{ {L^2(Q)}}^{ \frac{1}{p}},$ we obtain $\frac{d (\phi _{2p}(t))^{2p} }{dt }\le - c (\phi _{2p}(t))^{2p+\frac{4p}{n}} \phi _{p}(t)^{-\frac{4p}{n}},$ hence for almost every $t\in J$ we have

$$\begin{aligned} \frac{d (\phi _{2p}(t))^{-4p/n} }{dt } \ge \frac{2 c}{n}\Big (\phi _{p}(t)\Big )^{-\frac{4p}{n}}, \end{aligned}$$

the conclusion follows after integrating since $\phi _p(\cdot )$ is absolutely continuous. $\square$

We choose now $p=2^l, l\ge 1,$ we define inductively the functions

$$\begin{aligned} f_2=\Vert u_0\Vert _{L^2(Q)} \end{aligned}$$

and

$$\begin{aligned} f_{2p}= (2/l)^{-n/4p}(p-1)^{n/2p} f_p. \end{aligned}$$

By proceeding as in Lemma 5.6 and 5.7 in [10], we are led to the following results.

Lemma 4.6

For all p of the form $2^l, l\ge 1$

$$\begin{aligned} \phi _p(t)\le c t^{-\frac{n}{2}(\frac{1}{2}-\frac{1}{p})}f_p \end{aligned}$$

(4.6)

where c is a positive constant depending on n.

Lemma 4.7

Let u(t) be the weak solution of (3.20). Then for all $t>0$, we have that

$$\begin{aligned} \Vert u(t)\Vert _{L^\infty (Q)}\le c t^{-\frac{n}{4}} \Vert u_0\Vert _{L^2(Q)}, \end{aligned}$$

(4.7)

c is a positive constant.

We are now in position to prove Theorem 4.4.

Proof

From the above lemma, it follows that $\Vert U(t,s)u_0\Vert _{L^\infty (Q)}\le c (t-s)^{-\frac{n}{4}} \Vert u_0\Vert _{L^2(Q)},$ hence $\Vert U(t,s)\Vert _{\mathcal {L}(L^2\rightarrow L^\infty )}\le c (t-s)^{-\frac{n}{4}}$ since the initial time we choose is not relevant. $\square$

Hence from the Riesz–Thorin Theorem, we can prove.

Proposition 4.8

$$\begin{aligned} \Vert U(t,s)\Vert _{\mathcal {L}(L^2\rightarrow L^{2p})} \le (c (t-s)^{-\frac{n}{4}})^{1-\frac{1}{p}}. \end{aligned}$$

We next prove the ultracontractivity property.

Theorem 4.9

For all $p\in [1,2)$, there exists an operator $U_p(t,s)\in \mathcal {L}(L^p(Q))$ which extends U(t, s) and satisfies

$$\begin{aligned} \Vert U_p(t,s)\Vert _{\mathcal {L}(L^p(Q))}\le 1. \end{aligned}$$

Moreover, U(t, s) is ultracontractive: there exists a $c\in \mathbb {R}^+$ depending on n such that

$$\begin{aligned} \Vert U_1(t,s)u(s)\Vert _{L^\infty (Q)}\le c (t-s)^{-\frac{n}{2}} \Vert u(s)\Vert _{L^1(Q)} \end{aligned}$$

for every $u\in L^1(Q)$ and $0\le s< t.$

Proof

As $E(t,v,u)=E(t,u,v),$ the evolution operators associated with the two forms coincide with U(t, s). From (2.22) in [10], we have that the adjoint $U(t,s)^\prime = U(T-s,T-t),$ for all $0\le s\le t\le T.$ For $p\in [1,2)$, we define

$$\begin{aligned} U_p(t,s)= (U_{p'}(T-s,T-t))^\prime \end{aligned}$$

and we note that $U_2(t,s)=U(t,s)$ on $L^2(Q),$ that is $U_p(t,s)$ is an extension of U(t, s). Moreover, since the norm of the adjoint operators in Banach spaces is the same, $U_p(t,s)$ is a contraction from $L^p(Q)$ to $L^p(Q)$ also for $p\in [1,2).$

We now compute

$$\begin{aligned}{} & {} \Vert U_1(t,s)\Vert _{\mathcal {L}{(L^1\rightarrow L^2)}}=\Vert (U_\infty (T-s,T-t))^\prime \Vert _{\mathcal {L}{(L^1\rightarrow L^2)}}= \\{} & {} \Vert U_\infty (T-s,T-t)\Vert _{\mathcal {L}{(L^2\rightarrow L^\infty )}}=\Vert U(T-s,T-t)\Vert _{\mathcal {L}{(L^2\rightarrow L^\infty )}} \\{} & {} \quad \le c (t-s)^{-\frac{n}{4}} \end{aligned}$$

where the last inequality follows from Theorem 4.4.

Now from 2, in Theorem 3.8, we have

$$\begin{aligned}{} & {} \Vert U_1(t,s)\Vert _{\mathcal {L}{(L^1\rightarrow L^\infty )}}= \Vert U_1(t,\frac{t+s}{2})U_1(\frac{t+s}{2}, s)\Vert _{\mathcal {L}{(L^1\rightarrow L^\infty )}} \\{} & {} \quad \le \Vert U(t,\frac{t+s}{2})\Vert _{\mathcal {L}{(L^2\rightarrow L^\infty )}} \Vert U_1(\frac{t+s}{2}, s)\Vert _{\mathcal {L}{(L^1\rightarrow L^ 2)}} \\{} & {} \quad \le c (t-s)^{-\frac{n}{2}}. \end{aligned}$$

$\square$

Theorem 4.10

Under the hypothesis of Theorem 3.8, the evolution operator U(t, s) associated with the family A(t) satisfies the following properties.

• 1. For every $\theta : 0\le \theta <\eta +\frac{1}{2}$ and $0\le s<t\le T,$

  $$\begin{aligned} \mathcal {R}(U(t,s)) \subset \mathcal {D}(A(t)^\theta ). \end{aligned}$$
• 2.
  $$\begin{aligned} \Vert A(t)^\theta U(t,s)\Vert _{\mathcal {L}(L^{2})}\le c_\theta (t-s)^{-\theta }, \; 0\le s<t\le T. \end{aligned}$$

• 3. For $0<\beta< \gamma < \eta +\frac{1}{2}$ $\Vert A(t)^\gamma U(t,s) A(s)^{-\beta }\Vert _{\mathcal {L}(L^{2})} \le c_\gamma (t-s)^{\beta -\gamma }.$

• 4.
  $$\begin{aligned} \Vert [U(t+\tau ,t)-U(t,t)] A(t)^{-\beta }\Vert _{\mathcal {L}(L^{2})}\le c \tau ^\beta ,\;\;\; \tau >0, 0<\beta <1. \end{aligned}$$

Proof

For 1 to 3, see Section 8.1 in [35]. We prove 4. From Theorem 3.8, for every $\epsilon >0$ we have

$$\begin{aligned}{} & {} \Vert [U(t+\tau ,t)-U(t+\epsilon ,t)] A(t)^{-\beta }\Vert _{\mathcal {L}(L^{2})} =\Vert \int _{t+\epsilon }^{t+\tau } \frac{\partial U(\sigma ,t)}{\partial \sigma } A(t)^{-\beta } d\sigma \Vert _{\mathcal {L}(L^{2}(Q))}= \\{} & {} \Vert \int _{t+\epsilon }^{t+\tau } A(\sigma )U(\sigma ,t)A(t)^{-\beta } d\sigma \Vert _{\mathcal {L}(L^{2}(Q))} \le \int _{\epsilon }^{\tau }\Vert A(\sigma )U(\sigma ,t)A(t)^{-\beta }\Vert _{\mathcal {L}(L^{2}(Q))} d\sigma \\{} & {} \quad \le c \int _{t+\epsilon }^{t+\tau }|\sigma -t|^{\beta -1}d\sigma = c/\beta (\tau ^\beta -\epsilon ^\beta ). \end{aligned}$$

By 3, the conclusion follows by passing to the limit as $\epsilon \rightarrow 0$ and taking into account that U(t, s) is strongly continuous. $\square$

Remark 4.11

The above properties still hold for the family of evolution operators extended to $L^p(Q).$

5 The semilinear problem

We recall the properties of the abstract non-homogeneous Cauchy problem.

$$\begin{aligned} \left\{ \begin{array}{ll} u_t=A(t)u(t)+ f(t), &{} \hbox {for}\;\, 0<t\le T \\ u(0)=\phi , &{} \hbox {} \end{array} \right. \end{aligned}$$

(5.1)

where A(t) satisfies Theorem 3.7, $\phi \in L^2(Q)$ and $f\in C^{0,\delta }([0,T], L^2(Q))$

Theorem 5.1

For every $\phi \in L^2(Q)$ and $f\in C^{0,\delta }([0,T], L^2(Q))$ there exists a unique $u(t)\in C([0,T]; L^2(Q))\cap C^1((0,T]; L^2(Q)),$ $A(t)u\in C((0,T]; L^2(Q))$ such that (5.1) holds. Moreover,

$$\begin{aligned} \Vert u(t)\Vert _{L^2(Q))}{} & {} +t\Vert \frac{d u(t)}{dt}\Vert _{L^2(Q))} + t \Vert A(t)u(t)\Vert _{L^2(Q))}\\{} & {} \quad \le c (\Vert \phi \Vert _{L^2(Q))}+\Vert f\Vert _{C^{0,\delta }([0,T], L^2(Q))} ), \end{aligned}$$

for $0<t\le T,$ where c is a positive constant depending on c in (3.19).

Finally,

$$\begin{aligned} u(t)=U(t,0)\phi + \int _{0}^{t}U(t,\tau ) f(\tau )d\tau \end{aligned}$$

(see Theorem 3.9 in [35]).

We consider now the abstract semilinear Cauchy problem

$$\begin{aligned} (P)\qquad \left\{ \begin{array}{l} \frac{\textrm{d}u(t)}{\textrm{d}t} = A(t) u(t) + J(u(t)),\qquad 0\le t\le T\\ u(0)=\phi \end{array} \right. \end{aligned}$$

(5.2)

where $A(t):\mathcal {D} (A(t))\subset L^2(Q)\rightarrow L^2(Q)$ is the family of operators associated to the energy form E(t, u) introduced in (3.11), T is a fixed positive real number, $\phi$ is a given function in $L^2(Q)$. We assume that for every $t\in [0,T],$ J is a mapping from $L^{2p}(Q)\rightarrow L^2(Q),p>1$ locally Lipschitz, , i.e., Lipschitz on bounded sets in $L^{2p}(Q)$; we let ${l(r)}$ denote the Lipschitz constant of J:

$$\begin{aligned} \Vert J(u)-J(v)\Vert _{L^2(Q)} \le {l(r)} \Vert u-v\Vert _{L^{2p}(Q)} \end{aligned}$$

(5.3)

whenever $\Vert u\Vert _{L^{2p}(Q)}\le r,\Vert v\Vert _{L^{2p}(Q)}\le r$. We also assume that $J(0)=0$. This assumption is not necessary in all that follows, but it simplifies the calculations (see [33]). In order to prove the local existence theorem, we make the following assumption on the growth of ${l(r)}$ when $r\rightarrow \infty .$

$$\begin{aligned} \text{ Let }\; a:=\frac{n}{4}(1-\frac{1}{p}), \quad \text{ there } \text{ exists } \quad 0<b<a: \; {l(r)}= {\mathcal {O}}(r^\frac{1-a}{b}), r\rightarrow \infty ; \quad \quad \quad (g) \end{aligned}$$

we note that $0<a<1,$ for $n\le 4$ and $p>1.$ Let $p>1$. Following the approach in Theorem 2 in [33] and adapting the proof of Theorem 5.1 in [21], we have the following result.

Theorem 5.2

Let condition (g) hold. Let $K>0$ be sufficiently small, $\phi \in L^2(Q)$ and

$$\begin{aligned} \limsup _{t\rightarrow 0}\Vert t^b U(t,0)\phi \Vert _{L^{2p}(Q)}<K. \end{aligned}$$

(5.4)

There exists a $\overline{T} >0$ and a unique mild solution,

$$\begin{aligned} u\in C([0,\overline{T}],L^2(Q))\cap C((0,\overline{T}], L^{2p}(Q)) \end{aligned}$$

with $u(0)=\phi$ and $\Vert t^b u(t)\Vert _{L^{2p}(Q)}< 2K$ satisfying for every $t\in [0,\overline{T}]$,

$$\begin{aligned} u(t)= U(t,0) \phi +\int _0^t U(t,s) J(u(s)) ds \end{aligned}$$

(5.5)

with the integral being both an $L^2-$ valued and $L^{2p}-$valued Bochner integral.

The claim of the Theorem is proved by a contraction mapping argument on suitable spaces of continuous functions with values in Banach spaces. We adapt the proof of Theorem 5.1 in [21] to the new functional setting and for the reader’s convenience we sketch it.

Proof

Let Y be the complete metric space defined as follows,

$$\begin{aligned}{} & {} Y=\{u\in C([0,\overline{T}],L^2(Q))\cap C((0,\overline{T}], L^{2p}(Q)); \nonumber \\{} & {} \quad u(0)=\phi ; \Vert t^b u(t)\Vert _{L^{2p}(Q) }<2K \text{ for } \text{ all } \;t\in [0,\overline{T}]\} \end{aligned}$$

(5.6)

equipped with the metric

$$\begin{aligned} d(u,v)=max\{\Vert u-v\Vert _{C([0,\overline{T}],L^2(Q))}, \sup _{(0,\overline{T}]}t^b \Vert u(t)-v(t)\Vert _{L^{2p}(Q)}\}. \end{aligned}$$

For $w\in Y$, let $\mathcal {F}w(t)=U(t,0)\phi +\int _0^t U(t,s) J(w(s)) ds$. By using arguments similar to those used in the proof of Theorem 5.1 of [21], we can prove that for $u\in Y,$ $\mathcal {F}u\in C([0,T],L^2(Q))\cap C((0,T], L^{2p}(Q))$ and of course $\mathcal {F}u(0)=\phi$. We can prove that

$$\begin{aligned} \limsup _{t\rightarrow 0}\Vert t^b \mathcal {F}w(t)\Vert _{L^{2p}(Q)}<2K \; \text{ for } \text{ all }\; t\in [0,\overline{T}]. \end{aligned}$$

(5.7)

Hence, $\mathcal {F}:Y\rightarrow Y$, by choosing suitably $\overline{T}$ and K we can prove that it is a contraction. $\square$

Remark 5.3

If $J(u)= |u|^{p-1} u$ then ${l(r)}= {\mathcal {O}}(r^{p-1}), r\rightarrow \infty$ Thus condition (g) is satisfied for $b=\frac{1}{p-1}-\frac{n}{4p}$ with $p> 1 +\frac{4}{n}$;

From Proposition 4.10, we have $\mathcal {R}(U(t,s))\subset D(A(t)), \forall 0<s\le t$ and we can prove that the following regularity result holds (see also Theorem 5.3 in [21]).

Theorem 5.4

Make the assumptions of Theorem 5.2.

• (a) Let condition (g) hold, then the solution u(t) can be continuously extended to a maximal interval $(0,T_\phi )$ as a solution of (5.5), with $\Vert u(t)\Vert _{L^{2p}(Q)}< \infty$.

• (b)
  $$\begin{aligned}{} & {} u\in C([0,T_\phi ),L^2(Q))\cap C((0,T_\phi ), L^{2p}(Q))\cap C^1((0,T_\phi ),L^2(Q)) \\{} & {} Au(t)\in \;C((0,T_\phi );\;L^2(Q)) \end{aligned}$$

  and satisfies

  $$\begin{aligned}{} & {} \qquad \qquad \qquad \frac{\textrm{d}u(t)}{\textrm{d}t}=A(t)u(t)+J(u(t)), \;\text{ for } \text{ every }\;\; t \in (0,T_\phi ), \quad \quad \quad \quad \qquad u(0)=\phi . \end{aligned}$$

That is, it is a classical solution.

Proof

As to the proof of condition a), we follow Theorem 2 in [33]. From the proof of Theorem 5.2, it turns out that the minimum existence time for the solution to the integral equation is as long as $\Vert t^b U(t,s)\phi \Vert _{L^{2p}(Q)} \le K,$ (see also Corollary 2.1. in [33]).

To prove that the mild solution is classical, we use the classical regularity results for linear equations, see Theorem 5.1, by proving that $J(u)\in C^{0,\delta }( (0,T],L^2(Q))$ for any fixed $T<T_\phi .$

Taking into account the local Lipschitz continuity of J(u), it is enough to show that u(t) is Hölder continuous from $(\epsilon , T)\; \forall \epsilon >0$ into $L^{2p}$. Let $\psi = u(\epsilon ),$ we set $w(t)= U(t,0)\psi +\int _0^t U(t,s) J(w(s)) ds,$ if we prove that

$$\begin{aligned} w(t)\in C^0([0,T];\;L^{2p}(Q))\cap C^1([0,T]), L^2(Q)) \end{aligned}$$

and

$$\begin{aligned} A(t)w\;\in C([0,T];\;L^2(Q)) \end{aligned}$$

then, as $u(t+\epsilon )= w(t)$ due to the uniqueness of the solution of (5.5), we deduce that

$$\begin{aligned} u(t) \in C^1([\epsilon ,T+\epsilon );\;L^2(Q))\cap C([ \epsilon ,T+\epsilon ), L^{2p}(Q)) \end{aligned}$$

and,

$$\begin{aligned} A(t)u(t)\in \;C([ \epsilon ,T+\epsilon );\;L^2(Q)) ), \end{aligned}$$

for every $\epsilon >0,$ hence u(t) is a classical solution (see claim b). Let $\sup _{t\in (0,T)}\Vert w\Vert _{L^{2p}(Q)}\le r.$ Since U(t, 0) is differentiable in $(\epsilon , T)$ then it is Hölder continuous for any exponent $\gamma \in (0,1).$ We now prove that

$$\begin{aligned} v(t)=\int _0^t U(t,s) J(w(s)) ds \end{aligned}$$

is Hölder continuous too. Let $0\le t\le t+\tau \le T,$

$$\begin{aligned}{} & {} v(t+\tau )-v(t)=\int _{0}^{t+\tau }U(t+\tau ,s)J(w(s))ds-\int _{0}^{t} U(t,s)J(w(s))ds= \\{} & {} \int _{0}^{t}(U(t+\tau ,s)-U(t,s))J(w(s))ds+\int _{t}^{t+\tau }U(t+\tau ,s)J(w(s))ds:=v_1(t)+v_2(t). \end{aligned}$$

Hence,

$$\begin{aligned}{} & {} \Vert v_1(t)\Vert _{L^{2p}(Q)}\le \int _{0}^{t}\Vert (U(t+\tau ,t)U(t,s)-U(t,s)) J(w(s))\Vert _{L^{2p}(Q)}ds= \\{} & {} \int _{0}^{t}\Vert (U(t+\tau ,t)-I)A(t)^{-\beta }A(t)^{\beta }U(t,s)) J(w(s))\Vert _{L^{2p}(Q)}= \\{} & {} \int _{0}^{t}(\Vert \int _{t}^{t+\tau }A(\sigma )U(\sigma ,t)A(t)^{-\beta } d\sigma ) A(t)^{\beta }U(t,s)) J(w(s))\Vert _{L^{2p}(Q)}ds \\{} & {} \quad \le \int _{0}^{t}(\int _{t}^{t+\tau } \Vert A(\sigma )U(\sigma ,t)A(t)^{-\beta }\Vert _{\mathcal {L}(L^{2p}(Q))} d\sigma ) \Vert A(t)^{\beta }U(t,\frac{s+t}{2})U(\frac{s+t}{2}, s)\\{} & {} \quad J(w(s))\Vert _{L^{2p}(Q)} ds. \end{aligned}$$

Taking into account 2 and 3 of Theorem 4.10

$$\begin{aligned}{} & {} \le \int _{0}^{t}( \int _{t}^{t+\tau } c|\sigma -t|^{\beta -1} d\sigma ) \Vert A^\beta (t)U(t,\frac{s+t}{2})\Vert _{\mathcal {L}(L^{2p})} \Vert U(\frac{s+t}{2}, s)J(w(s))\Vert _{L^{2p}(Q)} ds \\{} & {} \le \int _{0}^{t} \frac{\tau ^\beta }{\beta } (\frac{t-s}{2})^{-\beta } c\Big (\frac{t-s}{2}\Big )^{\frac{-n}{4}(1-\frac{1}{p})} \Vert J(w(s))\Vert _{L^2(Q)} ds \\{} & {} \le c \int _{0}^{t} \frac{\tau ^\beta }{\beta } (\frac{t-s}{2})^{-\beta }(\frac{t-s}{2})^{-a} {l(r)} r ds. \end{aligned}$$

If we choose $\beta <1-a,$ we obtain $\Vert v_1(t)\Vert _{L^{2p}(Q)}\le c\tau ^\beta .$

As to the function $v_2$, we have

$$\begin{aligned}{} & {} \Vert v_2(t)\Vert _{L^{2p}(Q)}\le \int _t^{t+\tau }\Vert U(t+\tau ,s)J(w(s))\Vert _{L^{2p}(Q)}ds \\{} & {} \quad =\int _{t}^{t+\tau } \Vert U(t+\tau ,s)\Vert _{\mathcal {L}(L^2\rightarrow L^{2p})}\Vert J(w(s))\Vert _{L^2(Q)}ds \\{} & {} \quad \le (c)^{1-1/p} \frac{\tau ^{1-a}}{1-a} {l(r)} r = c \tau ^{1-a}. \end{aligned}$$

Therefore if $\beta <1-a,$ v(t) is Hőlder continuous on [0, T] with exponent $\beta$. $\square$

5.1 Global existence theorem

We now give a sufficient condition on the initial datum in order to obtain a global solution adapting Theorem 3 (b) in [34].

Theorem 5.5

Let condition g) hold. Let $q= \frac{2np}{n+4pb},$ $\phi \in L^q(Q),$ and $\Vert \phi \Vert _{L^q(Q)}$ be sufficiently small, then there exists a $u\in C([0,\infty ), L^q(Q))$ which is a global solution of (5.5).

Proof

Since $q<2p$, from Proposition 4.8 it follows that U(t, s) is a bounded operator from $L^q$ into $L^{2p}$ with

$$\begin{aligned} \Vert U(t,s)\Vert _{L^q\rightarrow L^{2p}} \le M (t-s)^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{2p})}\equiv M (t-s)^{-b}, \end{aligned}$$

hence

$$\begin{aligned} \Vert t^b U(t,s) \phi \Vert _{L^{2p}(Q)}\le M \Vert \phi \Vert _{L^q(Q)}; \end{aligned}$$

by choosing $\Vert \phi \Vert _{L^q(Q)}$ sufficiently small from Theorem 5.2 there exists a local solution of (5.5), $u \in C([0,T], L^q(Q))$. Furthermore, $u \in C((0,T],L^{2p}(Q))$ and $\Vert t^b u(t) \Vert _{L^{2p}}\le 2\,M \Vert \phi \Vert _{L^q(Q)}$ (See claim 1) in Theorem 5.2 (a)). From Theorem 5.4 (a) to show that u(t) is a global solution, it is enough to show that $\Vert u(t) \Vert _{L^{2p}(Q)}$ is bounded for every $t>0.$ We will prove that $\Vert t^b u(t) \Vert _{L^{2p}(Q)}$ is bounded for every $t >0,$ and we will use the notations of the proof in Theorem 5.2. We choose N such that $l(r)\le N r^{\frac{1-a}{b}}, r\ge 1$. Then,

$$\begin{aligned}{} & {} \Vert t^b u(t)\Vert _{L^{2p}(Q)}\le M\Vert \phi \Vert _{L^q(Q)} + t^b \int _0^t \Vert U(t,s)\Vert _{L^2\rightarrow L^{2p}}\Vert J(u(s))\Vert _{L^2(Q)} ds \\{} & {} \quad \le M\Vert \phi \Vert _{L^q(Q)}+ M N (2M \Vert \phi \Vert _{L^q(Q)} )^{\frac{1-a}{b}} t^b \int _0^t (t-s)^{-a} s^{a-1-b}\Vert s^b u(s)\Vert _{L^{2p}(Q)} ds \\{} & {} \quad \le M\Vert \phi \Vert _{L^q(Q)} +NM (2M \Vert \phi \Vert _{L^q(Q)})^{\frac{1-a}{b}}\sup _{t\in [0,T]} \Vert t^b u(t)\Vert _{L^{2p}(Q)} \int _0^1 (1-s)^{-a} s^{a-1-b} ds. \end{aligned}$$

Let $f(T)= \sup _{t\in [0,T]} \Vert t^b u(t)\Vert _{L^{2p}(Q)},$ f(T) is a continuous nondecreasing function with $f(0)=0,$ which satisfies

$$\begin{aligned} f(T) \le M \Vert \phi \Vert _{L^q(Q)} + (2M \Vert \phi \Vert _{L^q(Q)})^{\frac{1-a}{b}} N M B f(T), \end{aligned}$$

if $M \Vert \phi \Vert _{L^q(Q)}\le \alpha$ and $2^{\frac{1-a}{b}} NMB \alpha ^{\frac{1-a}{b}}<1.$ Then, f(T) can never equal $2\alpha .$ If it did we would have $2\alpha \le \alpha + ( 2 \alpha )^{\frac{1-a}{b}} NBM$, i.e., $\alpha \le ( 2 \alpha )^{\frac{1-a}{b}} NBM$ which is false if $\alpha >0$ is small enough. This proves that for $\Vert \phi \Vert _{L^q(Q)}$ sufficiently small, $\Vert t^b u(t)\Vert _{L^{2p}(Q)}$ must remain bounded and the claim follows. $\square$