Global solutions of abstract semilinear parabolic equations with memory terms

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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 equa- tions 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

˙u(t) = Au(t) + F (u(t)) + g(t) t≥ 0, u(0) = u0 ∈ X ,

(1.1)

where X is a real Hilbert space, A : D(A) ⊂ X → X is a self-adjoint, strictly negative linear operator on X, F is a nonlinear X-valued map defined on the domain of (−A)12, and g is an integrable function.

A natural generalization of the above problem is the integro-differential equation

˙u(t) + Z t

0 α(t− s) ˙u(s)ds = Au(t) +Z t

0 β(t− s)Au(s)ds + F (u(t)) + g(t) t≥ 0, u(0) = u0,

(1.2)

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: <cannarsa@axp.mat.uniroma2.it>

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Via A. Scarpa 16, 00161 Roma (Italy);

e-mail: <sforza@dmmm.uniroma1.it>

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As usual in the theory of evolution equations, we first look at the linear problem obtained taking F ≡ 0 in (1.2), that is

˙u(t) + Z t

0

α(t− s) ˙u(s)ds = Au(t) + Z t

0

β(t− s)Au(s)ds + g(t) t≥ 0, u(0) = u0,

(1.3)

The solution of the above problem, for u0 ∈ D(A) and g ∈ L2(0, T ; X), can be constructed ap- plying 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 T

0 hα ∗ ˙u, uid τ ≥ h(α ∗ u)(T ), u(0)i −Z T

0

α(τ )d τku(0)k2 ∀u ∈ H1(0, T ; X) , (1.4) that holds under an extra assumption on α (see also [13]). In sum, we show that, for any u0 ∈ X and g ∈ L1(0, T ; X), (1.3) has a unique solution u∈ C([0, T ]; X) ∩ L2(0, T ; D((−A)12)) . Incidentally, we note that the L2(0, T ; D((−A)12)) 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

u(t) = S(t)u0+ Z t

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 conditionkF (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

hF (x), xi ≤ εk(−A)12xk2+ Cε(1 +kxk)L(kxk) ∀x ∈ D((−A)12) , (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

0≤t≤τku(t)k2+ Z τ

0 k(−A)12u(r)k2dr≤ C1 τ ∈ [0, T [,

where u is the solution of (1.2) on [0, τ ] and C1is 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 u0 ∈ D((−A)12) and g ∈ L2(0, T ; X), we obtain a maximal regularity result for u, namely that

u∈ H1(0, T ; X)∩ C([0, T ]; D((−A)12))∩ L2(0, T ; D(A)) . (1.7)

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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

˙u(t) + Z t

−∞α(t− s) ˙u(s) ds = Au(t) + Z t

−∞β(t− s)Au(s) ds + F (u(t)) + h(t) t≥ 0, u(t) = v(t), 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 H1(−∞, 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

a0∂u

∂t(t, ξ) + Z t

−∞a(t− s)∂u

∂t(s, ξ)ds = (1.9)

= b04u(t, ξ) + Z t

−∞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 litera- ture 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 mod- els 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 inter- esting 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 consid- ered 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.

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