Global solutions of abstract semilinear parabolic

Global solutions of abstract semilinear
parabolic equations with memory terms
Piermarco Cannarsa∗
Daniela Sforza†
Abstract
The main purpose of this paper is to obtain the existence of global solutions to semilinear
integro-differential equations in Hilbert spaces for rather general convolution kernels and
nonlinear terms with superlinear growth at infinity. The included application to a nonlinear
model of heat flow in materials of fading memory type provides motivations for the abstract
theory.
1
Introduction
The existence, uniqueness and asymptotic behaviour of solutions to semilinear evolution equations is a topic that has been extensively studied in research papers and, nowadays, is also
treated in many textbooks. For instance, the monographs [17, 20, 25] contain a comprehensive
survey of introductory—as well as advanced—results for the Cauchy problem


˙
= Au(t) + F (u(t)) + g(t)
 u(t)
t ≥ 0,
(1.1)

 u(0) = u ∈ X ,
0
where X is a real Hilbert space, A : D(A) ⊂ X → X is a self-adjoint, strictly negative linear
1
operator on X, F is a nonlinear X-valued map defined on the domain of (−A) 2 , and g is an
integrable function.
A natural generalization of the above problem is the integro-differential equation
Z t
Z t



˙ +
α(t − s)u(s)ds
˙
= Au(t) +
β(t − s)Au(s)ds + F (u(t)) + g(t)
 u(t)



0
0
t ≥ 0,
(1.2)
u(0) = u0 ,
where α and β are given integrable functions on [0, +∞[. This paper will mainly focus on the
well-posedness of such a problem. Before explaining the key-points of our analysis, let us point
out that our main goal is to obtain a global existence result for (1.2) when α and β are just
L1 (0, +∞) functions and F (u) has superlinear growth at infinity.
∗
Dipartimento di Matematica, Universit`
a di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma
(Italy); e-mail: <[email protected]>
†
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Via A. Scarpa 16, 00161 Roma (Italy);
e-mail: <[email protected]>
1
As usual in the theory of evolution equations, we first look at the linear problem obtained
taking F ≡ 0 in (1.2), that is
Z t
Z t



˙ +
α(t − s)u(s)ds
˙
= Au(t) +
β(t − s)Au(s)ds + g(t)
 u(t)



0
0
t ≥ 0,
(1.3)
u(0) = u0 ,
The solution of the above problem, for u0 ∈ D(A) and g ∈ L2 (0, T ; X), can be constructed applying a result by Pr¨
uss for linear integral equations (see [26, Theorem 8.7]). We want, however,
to solve the problem for u0 ∈ X and g ∈ L1 (0, T ; X). This can be done approximating u0 by
more regular initial conditions, provided one can prove uniform bounds for the corresponding
solutions. Such bounds can in turn be obtained by the standard multiplier method using the
coercivity estimate
Z
0
T
hα ∗ u,
˙ uid τ ≥ h(α ∗ u)(T ), u(0)i −
Z
T
0
α(τ )d τ ku(0)k2
∀u ∈ H 1 (0, T ; X) ,
(1.4)
that holds under an extra assumption on α (see also [13]). In sum, we show that, for any
1
u0 ∈ X and g ∈ L1 (0, T ; X), (1.3) has a unique solution u ∈ C([0, T ]; X) ∩ L2 (0, T ; D((−A) 2 )) .
1
Incidentally, we note that the L2 (0, T ; D((−A) 2 )) regularity of u for g ∈ L1 (0, T ; X) seems to
be a new result even for the non-integral case (1.1).
Using the solution of (1.3) with g = 0—briefly, the resolvent of (1.3)—that we denote by
S(t)u0 , we define the mild solution of (1.2), with u0 ∈ X and g ∈ L1 (0, T ; X), as the solution of
the equation
Z
t
u(t) = S(t)u0 +
0
(S − % ∗ S)(t − s)[F (u(s)) + g(s)] ds,
(1.5)
where % ∈ L1loc (0, +∞) satisfies, in turn, the integral equation % + α ∗ % = α . Then, in order to
show that the mild solution of (1.2) is global, the only information we need is an appropriate a
priori estimate for u.
Returning to problem (1.1), we recall that a typical assumption used to obtain a priori
bounds for solutions is the sublinear growth condition kF (x)k ≤ C(1 + kxk) or, more generally,
the one-sided condition hF (x), xi ≤ C(1 + kxk2 ). Here, we want to relax such an assumption
allowing superlinear growth of kF (x)k at infinity. For this purpose, following the approach of
[3], we shall assume that for any ε > 0 a constant Cε > 0 exists such that
1
hF (x), xi ≤ εk(−A) 2 xk2 + Cε (1 + kxk)L(kxk)
1
∀x ∈ D((−A) 2 ) ,
(1.6)
where
L(t) := (1 + |t|) log(e + |t|) log log(ee + |t|) . . .
(t ∈ R)
is the infinite product of iterated logarithms introduced in [6]. Combined with the multiplier
method and the lower bound (1.4), the above condition yields the a priori estimate
sup ku(t)k2 +
0≤t≤τ
Z
0
τ
1
k(−A) 2 u(r)k2 dr ≤ C1
τ ∈ [0, T [,
where u is the solution of (1.2) on [0, τ ] and C1 is independent of τ . This is exactly the inequality
we need to show that the solution is global, or that τ = T . Moreover, for smooth data, say
1
u0 ∈ D((−A) 2 ) and g ∈ L2 (0, T ; X), we obtain a maximal regularity result for u, namely that
1
u ∈ H 1 (0, T ; X) ∩ C([0, T ]; D((−A) 2 )) ∩ L2 (0, T ; D(A)) .
2
(1.7)
The last result also implies that the equation in (1.2) is satisfied almost everywhere in [0, T ].
Besides the interest in itself, another reason for studying (1.2) is that by solving this problem
we can also treat the history value problem
Z



˙ +
 u(t)



t
−∞
u(t) = v(t),
α(t − s)u(s)
˙
ds = Au(t) +
Z
t
−∞
β(t − s)Au(s) ds + F (u(t)) + h(t)
t ≥ 0,
t ≤ 0.
(1.8)
In fact, we observe that our global existence result for the Cauchy problem (1.2), together with
the maximal regularity (1.7), yields a similar result for (1.8) provided the history v belongs to
H 1 (−∞, 0; X) ∩ L2 (−∞, 0; D(A)), see Theorem 4.9.
It is well-known that the last problem can be used to describe physical phenomena, such
as the heat flow in materials for which the effects of memory cannot be neglected, see, e.g.,
[16, 23, 24]. A model problem for such a flow is the following
∂u
a0 (t, ξ) +
∂t
Z
t
−∞
a(t − s)
= b0 4u(t, ξ) +
Z
t
−∞
∂u
(s, ξ)ds =
∂t
(1.9)
b(t − s)4u(s, ξ)ds + f (u(t, ξ)) + h(t, ξ)
where t ≤ T and ξ ∈ Ω, Ω being a bounded open domain in RN with smooth boundary. The
results of this paper can be applied to (1.9) taking X = L2 (Ω) and F equal to the composition
operator
F (x)(ξ) = f (x(ξ))
(ξ ∈ Ω, x ∈ X) .
A natural growth condition for f to ensure the validity of our crucial assumption (1.6) is
tf (t) ≤ c(1 + t2 ) log(e + |t|) log log(ee + |t|) . . .
∀t ∈ R .
(1.10)
We note such a condition is very close to being optimal for the existence of global solutions, see
Remark 5.2.
To conclude this introduction some bibliographical comments are in order. Since the literature on integro-differential equations is huge, as one can see consulting the monographs [15, 26]
and the references therein, we will just recall some of the closest contributions to the topics
treated in the paper, with no pretensions to being exhaustive. For obvious reasons, linear models are the most studied in the literature. A work that certainly has strong connections with the
present set-up is the one by Giorgi and Gentili [13] that investigates parabolic integro-differential
equations from a different viewpoint, without aiming at maximal regularity. On the other hand,
fewer results are available for the existence of global solutions in the semilinear case. An interesting result is obtained in [14], where, however, attention is focussed on asymptotic behaviour
rather than global existence. Additional papers studying nonlinear integro-differential problems
are [1, 2, 8, 9, 21, 22, 27]. Superlinear growth conditions of logarithmic type were first considered by Cazenave and Haraux in [7], where the global existence of solutions with finite energy is
proved for semilinear wave equations. Such conditions were used to construct global solutions
of the non-integral Cauchy problem (1.1) in [3], and to study exact controllability in [4, 12].
This paper is organized in the following way. Section 2 contains notations and preliminaries
that will be used throughout. Section 3 is mainly devoted to the construction of the resolvent for
linear equations. In Section 4 we obtain global existence and maximal regularity for semilinear
problems. As an application of these results, in Section 5 we discuss the existence in the large
of solutions to parabolic integro-differential equations.
3
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