BLOW-UP OF SOLUTIONS OF A SEMILINEAR HEAT EQUATION WITH A MEMORY TERM

where u is the temperature, d is the diffusion coefficient, and the integral term represents the memory effect in the material. The study of this type of equations has drawn a considerable attention, see [3, 4, 10, 12, 13]. From a mathematical point of view, one would expect the integral term to be dominated by the leading term in the equation. Therefore, the theory of parabolic equations applies to this type of equations. In the absence of the memory term (g = 0), problem (1.1) has been studied by various authors and several results concerning global and nonglobal existence have been established. For instance, in the early 1970s, Levine [6] introduced the concavity method and

This equation arises from a variety of mathematical models in engineering and physical sciences.For example, in the study of heat conduction in materials with memory, the classical Fourier's law of heat flux is replaced by the following form: where u is the temperature, d is the diffusion coefficient, and the integral term represents the memory effect in the material.The study of this type of equations has drawn a considerable attention, see [3,4,10,12,13].From a mathematical point of view, one would expect the integral term to be dominated by the leading term in the equation.Therefore, the theory of parabolic equations applies to this type of equations.
In the absence of the memory term (g = 0), problem (1.1) has been studied by various authors and several results concerning global and nonglobal existence have been established.For instance, in the early 1970s, Levine [6] introduced the concavity method and showed that solutions with negative energy blow up in finite time.Later, this method was improved by Kalantarov and Ladyzhenskaya [5] to accommodate more general situations.Ball [2] also studied (1.1) with f (u,∇u) instead of |u| p−2 u and established a nonglobal existence result in bounded domains.This result had been extended to unbounded domains by Alfonsi and Weissler [1].
For the quasilinear case, Junning in [14] studied and established a global existence result.He also proved a nonglobal existence result under the condition where F(u) = u 0 f (s)ds.More precisely he showed that if there exists T > 0, for which (1.2) holds, then the solution blows up in a time less than T.This type of results have been extensively generalized and improved by Levine, Park, and Serrin in [7], where the authors proved some global, as well as nonglobal, existence theorems.Their result, when applied to problem (1.3), requires that We note that the inequality (1.5) implies (1.4).In a note, Messaoudi [8] extended the blow-up result to a solution with an initial datum satisfying In the present work, we consider (1.1) and show that, for suitable conditions on p and g, the blow-up can be obtained even for some solutions with positive initial energy.The present paper improves the one in [8] as it is only a special case.

Blow-up
In order to state and prove our result, we introduce the "modified" energy functional where For the relaxation function g and the number p, we assume that We also set where B = C * /l for C * the best constant of the Sobolev embedding H 1 0 (Ω) L p (Ω).By multiplying the equation in (1.1) by u t and integrating over Ω, we get, after some manipulations, see [9], for regular solutions.The same result can be established, for almost every t, by a simple density argument.
Similar to [11], we give a definition for a strong solution of (1.1).

Lemma 2.2. Let u be a strong solution of (1.1) with initial data satisfying
Then there exists a constant β > α such that ≥ β, (2.9) 90 Blow-up of solutions of a heat equation Proof.We first note that, by (2.1) and the Sobolev embedding, we have where 12) It is easy to verify that h is increasing for 0 where α is given in (2.8).Therefore, since E(0) < E 1 , there exists β > α such that h(β) = E(0).By using (2.11) we have which implies that ∇u 0 2 ≥ β.Now to establish (2.9), we suppose by contradiction that for some t 0 > 0 and, by the continuity of we can choose t 0 such that (2.17) Again the use of (2.11) leads to This is impossible since E(t) ≤ E(0), for all t ∈ [0,T).Hence (2.9) is established.