Singular kernel problems in materials with memory

Meccanica DOI 10.1007/s11012-014-0083-y ADVANCES IN DYNAMICS, STABILITY AND CONTROL OF MECHANICAL SYSTEMS Singular kernel problems in materials with memory Sandra Carillo Received: 14 April 2014 / Accepted: 24 November 2014  Springer Science+Business Media Dordrecht 2014 Abstract In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as well as the past history. Aiming to study new materials integrodifferential problems admitting singular kernels are compared. Specifically, on one side the temperature evolution in a rigid heat conductor with memory characterized by a heat flux relaxation function S. Carillo (&) Dipartimento di Scienze di Base e Applicate per l’Ingegneria - Sez. MATEMATICA, SAPIENZA Università di Roma, Via A. Scarpa 16, 00161 Rome, Italy e-mail: sandra.carillo@sbai.uniroma1.it; sandra.carillo@uniroma1.it singular at the origin, and, on the other, the displacement evolution within a viscoelastic model characterized by a relaxation modulus which is unbounded at the origin, are considered. One dimensional problems are examined; indeed, even if the results are valid also in three dimensional general cases, here the attention is focussed on pointing out analogies between the two different materials with memory under investigation. Notably, the method adopted has a wider interest since it can be applied in the cases of other evolution problems which are modeled by analogue integrodifferential equations. An initial boundary value problem with homogenous Neumann boundary conditions is studied. Keywords Viscoelasticity  Thermodynamics with memory  Integro-differential equations  Singular relaxation modulus  Singular kernel  Singular heat flux relaxation function 1 Introduction The models here considered are well known ones and refer to materials with memory as they are termed in the wide literature which is concerned about their physical (thermodynamical and/or mechanical) behaviours, on one side, and the many interesting analytical problems, on the other one. Specifically, in the case of a viscoelastic body, its deformation does 123 Meccanica not depend only on the mechanical status of the body in the present time, but, also, on its deformation history according to the well known model [1–3]. Viscoelastic components are more and more often used in devising new and smart materials in general. The interest in this subject is testified by many books and Conferences such as such as [4–7] which may be considered as sample ones; the first two are very recent books providing an overview on the subject aiming, the first one, to viscoelasticity models referring to study earthquakes and the second one to an updated overview on fractional calculus in linear viscoelasticity. Indeed, according to Fabrizio [8], who, in 2014, further to cite experimental results, analyzes the connection between Volterra and fractional derivatives models, the growing interest in models based on fractional derivatives is also due to the need to devise new tools to study materials whose behaviour cannot be described when the classical regularity hypothesis on the kernel in the integro-differential equations are assumed. This is the case studied, again in 2014, by Deseri et al. [9], who show how a fractional derivative model can be adapted to describe bio-materials. Indeed, the Special Issue of Discrete and Continuous Dynamical Systems—Series B dedicated to Mauro Fabrizio [10], comprises articles dedicated to materials with memory or which may be termed new, such as [11–24] and mathematical models to describe the behaviour of problems which arise in biological contexts [25–28]. These results motivate us to adopt, in this present article, less restrictive functional requirements on the kernel. The two books [6, 7] are Special Volumes which comprise results presented in Conferences devoted to new and smart materials together with analytical problems arising from the investigation of such models; these books are listed here as examples of the current interest in materials with memory and in the related mathematical models. The general regularity assumptions on the relaxation modulus guarantee the solution existence and uniqueness of Volterra type problems, as pointed out by many authors and firstly proved by Dafermos [29, 30]. Nevertheless, the idea of singular kernels to model particular cases of viscoelastic behaviours was introduced by Boltzmann [31] in the nineteenth century. Later, the same model was further investigated, since the middle of twentieth century, by Zimm et al. [1, 32–34] referring to polymers. On the other 123 hand, many authors [9, 35–38] pointed out also the applicative interest of new polymers and/or biomaterials whose mechanical response is not modeled by a Volterra type integro-differential equation with a regular kernel. A wide research activity is testified by many references such as [39–46] to mention some of those ones concerning singular kernel problems both under the analytical as well as the model point of view. Furthermore, other authors, such as Berti [47] and Grasselli and Lorenzi [48] study viscoelasticity problems exhibiting a singular memory kernel. The thermodynamical admissibility of a singular viscoelastic model characterized by a singular viscoelastic relaxation modulus is analyzed by Giorgi and Morro [49]. The references [50–58] are all concerning singular kernel problems both in rigid thermodynamics with memory as well as in viscoelasticity. Specifically, [50–52, 57, 58] study asymptotic behaviour of solutions. In this framework, the study here presented are part of a wide research project concerning the mechanical behavior of materials with memory, in which the author is involved. Thus, the aim here is to further develop results obtained in joint research works with Valente and Vergara Caffarelli [59–63]. Here, the attention is focussed on the existence and uniqueness of the solution admitted by singular kernel problems in materials with memory, previous and in progress results are comprised in [61–63]. Indeed, this study refers to Neumann boundary conditions while previous results were concerned about Dirichlet boundary conditions. The material is organized as follows. The opening Sect. 2 is concerned about the physical model. Crucial assumptions which characterize, in turn, the model of rigid heat conduction with memory and of isothermal viscoelasticity are comprised in the two sub-Sections. The key references, wherein the aspects of the models here of interest are given, are [64, 65] concerning rigid heat conduction and [2, 3] in the case of isothermal viscoelasticity. Notably, as pointed out in [66], the analogous functional spaces, wherein the solutions of the evolution problems are looked for, are obtained in the cases of the two different models. Indeed, throughout the whole article, the two different models are compared in connection to the singular kernel problems under investigation. In particular, analogies between the two models of materials with memory both under the physical as well as under the Meccanica functional spaces point of view are shown in [66], here the comparison is extended to singular kernel problems which were not considered previously. In particular, the analogy remains valid when the regularity requirement of the kernel is relaxed to consider also the case when, in turn, the heat flux relaxation function and the relaxation modulus are unbounded at the origin provided they are integrable (L1 ). The next Sections are concerned about evolution problems: Sects. 3 and 4, respectively, study evolution problems in heat conduction with memory and in isothermal viscoelasticity. Each Section is divided in two parts; the first one dedicated to classical regular problems, and, then, the second one, devoted to singular problems. Specifically, here singular problems with assigned initial and Neumann homogeneous boundary conditions are considered. In the subsequent Sect. 5 existence and uniqueness results are given. In the closing Sect. 6 some perspective problems and current investigations are mentioned. Detailed proofs of results previously stated are included in the Appendix. 2 The model of material with memory • no space dependence, i.e., x-dependence is omitted under the assumption that the material is homogeneous and isotropous. In particular, the approach presented by Fabrizio et al. [64], and, subsequently, in [65] is adopted. The internal energy e is assumed to be linearly related to the relative temperature u :¼ h  h0 ; where h0 denotes a fixed reference temperature, namely eðtÞ ¼ a0 uðtÞ: ð1Þ The heat flux q 2 IR3 ; when, in turn, g :¼ ru denotes the temperature-gradient, and Z t t ð2Þ g ðsÞ ¼ gðsÞds; ts the integrated history of the temperature-gradient, reads Z 1 kðsÞgðt  sÞds qðtÞ ¼  Z 10 _ gt ðsÞds: qðtÞ ¼ kðsÞ or ð3Þ 0 The heat flux relaxation function kðtÞ; in (3), is given by Z t _ ð4Þ kðtÞ ¼ k0 þ kðsÞds; 0 In this Section some of the key features of the model of a material with memory are recalled referring to the two cases under consideration, namely rigid heat conduction with memory and isothermal viscoelasticity. where k0  kð0Þ denotes the initial heat flux relaxation coefficient, that is the initial (positive) value assumed by the heat flux relaxation function. It is further required that 2.1 Rigid heat conduction with memory k_ 2 L1 ðIRþ Þ \ L2 ðIRþ Þ Here the model of a rigid heat conductor with memory, restricting only to a description of the physical assumptions and to some properties needed in the following, is briefly recalled. First of all, let X  R3 denote the body configuration, the main assumptions on the rigid heat conductor with memory are [64]: hence kð1Þ :¼ limt!1 kðtÞ ¼ 0: These assumptions imply the the material enjoys the fading memory property, namely, 8e [ 09~ a ¼ aðe; gt Þ 2 Rþ s:t:8a [ a~ Z 1 ð6Þ _ þ aÞ kðs gt ðsÞds \e; ) X is a connected set with a smooth boundary; X changes its thermodynamical status according to linear heat conduction with memory; that is, it depends on time via present and past times, i.e., on the thermal history of the material; The environment is assumed not to be affected by the presence of the body itself; which can be physically interpreted recalling that there is no heat flux when, at infinity, the thermal equilibrium is reached. The thermodynamical state of the conductor is characterized when, according to [64, 65, 67], the thermodynamic state function r: IR ! IR  IR3 which associates t ! rðtÞ  ðuðtÞ; gt Þ is given. Hence, the following vectorial space • • • and k 2 L1 ðIRþ Þ; ð5Þ 0 123 Meccanica  gt : ð0; 1Þ ! IR3 :  \1; 8s  0 ; C:¼ Z 1 • _ þ sÞ kðs gt ðsÞds 0 ð7Þ is introduced to characterize physically admissible thermodynamical phenomena, namely those ones associated to a finite heat flux. Note, that the condition which defines the function space C; can be written also in a different form since, corresponding to any arbitrary e [ 0, there exists a positive constant a~ ¼ aðe; gt Þ s.t. Z 1 _ þ aÞ kðs gt ðsÞds 0 ð8Þ Z 1 ¼ kðs þ aÞgðt  sÞds \e; 8a [ a~: 0 Remarkably, the introduction of the integrated history of the temperature-gradient is crucial to write the fading memory condition under a form which is analogous to the fading memory condition in the case of isothermal viscoelasticity, as shown in the next subSection, the state of the heat conductor is characterized when, according to [64, 65, 67], the thermodynamic state function r: IR ! IR  IR3 which associates t ! rðtÞ  ðhðtÞ; gt Þ is given. 2.2 Isothermal viscoelastic body with memory • • X changes its shape according to linear viscoelasticity; that is, its deformation depends on time via present and past times, i.e., the deformation history of the material; The environment is assumed not to be affected by the presence of the body itself; no space dependence, i.e., x-dependence is omitted under the assumption that the material satisfies both the isotropy and homogeneity conditions. Hence, [68, 69] the quantities which are needed to describe this model are the strain tensor EðtÞ; the stress tensor TðtÞ; the relaxation modulus GðtÞ l and its initial value, termed initial relaxation modulus Gð0Þ; l where t denotes the present time variable. The constitutive equation which characterizes a linear viscoelastic material is the classical Boltzmann–Volterra equation which relates the stress tensor TðtÞ 2 Sym to the strain history tensor E: ð1; t ! Sym : Z 1 _  sÞds or TðtÞ ¼ GðsÞ l Eðt 0 Z 1 ð9Þ _ TðtÞ ¼ Gl 0 EðtÞ þ GðsÞEðt l  sÞds; 0 where the fourth order tensor GðtÞ l denotes the elastic modulus, EðtÞ the value of the strain at the time t and Et the past history defined by Et : ð0; 1Þ ! Sym t ! Et ðsÞ :¼ Eðt  sÞ: The aim of this sub-Section is to briefly summarize the analytical description of the model of an isothermal viscoelastic material with memory. Indeed, the historical as well as phenomenological ideas which have been developed throughout the literature are far beyond the present study. An overview concerning isothermal viscoelasticity is comprised in [68] and in [69] as well as in references therein. Gentili [68] studies minimum free energy and its connection to maximum recoverable work, while Deseri et al. [69], are concerned about free energies, again in isothermal viscoelasticity with the aim to treat applications to partial differential equations. Here, first of all, a brief description of the essential properties which characterize the model of viscoelastic body [2, 41], are recalled. Denoted as X  R3 ; the body configuration, the key assumptions on the viscoelastic material are the following ones [2, 41]: • X is a connected set with a smooth boundary; 123 ð10Þ The elastic modulus is assumed such that its time derivative G l_ 2 L1 ðIRþ ; LinðSymÞÞ so that, for all positive t, Z t ð11Þ GðtÞ l ¼G l0þ GðsÞds; l_ G l 0 :¼ Gð0Þ; l 0 where the initial value of the elastic modulus Gl 0 is termed instantaneous elastic modulus [68]; furthermore, since Gl_ 2 L1 ; then Gð1Þ l :¼ lim GðtÞ l 2 LinðSymÞ ð12Þ t!1 which represents the equilibrium elastic modulus. The state and the strain history of a viscoelastic body is characterized, according to [2, 68, 69], by a viscoelastic state function r: IR ! Sym  Sym; which associates t7!rðtÞ:  ðEðtÞ; Et Þ: Hence, the viscoelastic state function is known when the strain tensor EðtÞ and the strain past history, Et which belong to a suitable Hilbert space, are assigned. Physically Meccanica meaningful viscoelastic phenomena, are characterized by a finite stress tensor TðtÞ for all times t and hence they belong to the vectorial space  Z 1 C :¼ Et : ð0; 1Þ ! Sym: Gðs l_ þ sÞEt ðsÞds 0  \1; 8s  0 : ð13Þ According to [68], the material is said to enjoy the fading memory property when, corresponding to any arbitrary e [ 0 there exists a positive constant a~ ¼ aðe; Et Þ s.t. Z 1 Gðs l_ þ aÞEðt  sÞds 0 Z 1 _  sÞds \e; 8a [ a~: ð14Þ ¼ Gðs l þ aÞEðt 0 The same way of reasoning followed in the previous sub-Section allows to establish that admissible states are only those ones related to a finite viscoelastic work, and the equivalence between any couple of different states as those ones associated to the same value of the viscoelastic work. Note the analogy between the two conditions (14) and (8): it shows that the couples, G; l Et ; on one side, and k; gt ; on the other one, play, respectively, the same role in the two different formulae. Remark Note that definition (14), and also (8), is not affected by the generalization here considered since when GðtÞ; l and also kðtÞ; is allowed to admit an _ is not integrable at infinite limit t ! 0; G; l_ and also k; the origin; however, it belongs to L1 ða; þ1Þ; 8a [ 0: 3 Heat conduction evolution problem In this Section the evolution problem of a rigid heat conductor with memory is considered. The classical regular initial boundary value problem with homogeneous initial and Neumann boundary conditions is recalled. In the next sub-Section the corresponding singular problem is studied. ut ¼ Z t kðsÞuxx ðx; t  sÞds þ f ðx; tÞ; ð15Þ 0 where in the term f, which represents the source term, also the past history of the material is incorporated. Generally, f is assumed to be sufficiently regular to allow integration and partial derivation with respect to time and, in addition, is also supposed to be L2 integrable in the space variable. This choice allows to write the corresponding evolution problem, when initial and boundary conditions are assigned, under the form of a Volterra equation. Here homogeneous Neumann boundary conditions ux ð0; tÞ ¼ ux ðL; tÞ ¼ 0; 8t [ 0; ð16Þ and the initial condition, at t ¼ 0 are imposed. The three-dimensional initial boundary value problem with homogeneous initial and Dirichlet boundary conditions is studied in [62]. Equation (15), on introduction of s :¼ t  s; reads Z t ð17Þ ut ¼ kðt  sÞuxx ðx; sÞds þ f ðx; tÞ: 0 Derivation with respect to time of the latter gives Z t _  sÞuxx ðsÞds þ f t ; ð18Þ kðt utt ¼ kð0Þuxx þ 0 then, consider the i.b.v.p. obtained imposing on (18) the following initial and boundary conditions ujt¼0 ¼ u0 ðxÞ; ut jt¼0 ¼ f ðx; 0Þ; ux joXð0;TÞ ¼ 0; t\T; ð19Þ where X denotes the interval ð0; LÞ; L [ 0: Note that the linear problem (18)–(19) is of the same form of the integro-differential problem which models the viscoelastic body evolution (26)–(27), in the next Section, hence, when the heat flux relaxation function k, finite at the origin, satisfies the thermodynamical assumptions (5), then Dafermos’ results [29, 30] imply the existence and uniqueness of the solution. 3.2 Singular memory kernel 3.1 Regular memory kernel The one-dimensional evolution equation which models rigid heat conduction with memory can be written In this sub-Section, the one-dimensional singular heat conduction problem is studied. Now, aiming to model a wider class of materials with memory, the functional 123 Meccanica requirements on the heat flux relaxation function k are relaxed, removing the condition k_ 2 L1 ð0; TÞ; thus _ kðtÞ kðtÞ [ 0; 0; €  0; kðtÞ PI;heat : uðtÞ ¼ t Z Kðt  sÞuxx ðsÞds þ u0 þ 0 Z t f ðsÞds; 0 ð25Þ t 2 ð0; 1Þ; ð20Þ is obtained. ð21Þ 4 Viscoelastic problem and 8T 2 IRþ : k 2 L1 ð0; TÞ \ C2 ð0; TÞ Now, the same method valid in studying the viscoelasticity problem, can be applied straightforwardly. Hence, key steps of the Approximation Strategy can be sketched as follows: • • • • construct suitable regular approximated problems; find approximated solutions ue ; 0\e 1; show the existence of u :¼ lime!0 ue ; prove the uniqueness of the limit solution u which represents a weak solution admitted by the singular problem. Accordingly, first of all the approximated problems are introduced: let ke ðÞ :¼ kðe þ Þ then, the problem PeD;heat can be defined Z t PeD;heat : uett ¼ ke ð0Þuexx þ k_e ðt  sÞuexx ðsÞds þ f : 0 ð22Þ The integro-differential problem (22), when conditions (19) are imposed on ue ; is regular since ke ð0Þ is finite and, therefore, [29, 30] admits a unique solution. It, in addition, corresponding to each value of e; is equivalent, to the following integral problem Z t Z t e e e e PI;heat : u ðtÞ ¼ K ðt  sÞuxx ðsÞds þ u0 þ f ðsÞds; 0 0 ð23Þ where, KðnÞ :¼ Z In this Section the linear integro-differential problem in the case of the one-dimensional viscoelastic classical model, is considered. Thus, from here on, the tensor G l is denoted as G since we are restricting our attention to the one-dimensional case. In the starting sub-Section, following the same lines as in the heat conduction problem, the classical regular case is recalled to point out the functional requirements the kernel satisfies in such a case. Then, in the subsequent sub-Section, the regularity requirements on G are relaxed. 4.1 Regular memory kernel According to Dafermos [29, 30], such a model can be represented by Z t _  sÞuxx ðsÞds þ f ; ð26Þ utt ¼ Gð0Þuxx þ Gðt 0 uð; 0Þ ¼ u0 ; ut ð; 0Þ ¼ u1 in X; ð27Þ where X ¼ ð0; LÞ; the initial and boundary conditions are assigned while, respectively, u and f denote the displacement and the external force which includes also the history of the material. In addition, according to the model assumptions (see, for instance [2, 41]) n kðsÞds Kð0Þ ¼ 0; ð24Þ GðtÞ [ 0; _ GðtÞ 0; €  0; GðtÞ 0 is well defined since k 2 L1 ð0; Þ; 8T 2 IRþ : K is termed integrated relaxation function. Partial derivation w.r.to t, twice, of (23) delivers (22) together with conditions (19). Note that, on use of (24), when e ¼ 0 and the superscripts 0 are omitted, the following well defined integral problem 123 ux ¼ 0 on R ¼ oX  ð0; TÞ; t 2 ð0; 1Þ; ð28Þ and, in addition, G_ 2 L1 ðIRþ Þ GðtÞ ¼ G0 Z t _ þ GðsÞds G1 :¼ lim GðtÞ; 0 t!1 ð29Þ Meccanica G enjoys the fading memory property (14). When the relaxation modulus satisfies assumptions (29), the problem (26)–(27), here termed Regular Memory Kernel, admits a unique solution according to Dafermos [29, 30]. 4.2 Singular memory kernel 0 In many applications, however, to model the physical behavior of new materials or polymers, the relaxation modulus does not satisfy the functional requirements (29). Hence, to model a wider class of materials with memory, the functional requirements imposed on G are relaxed, that is G is assumed to satisfy (28) further to the condition G 2 L1 ð0; TÞ \ C 2 ð0; TÞ 8T 2 IR: ð30Þ Note that, now, according to (28) and (30), the relaxation function GðtÞ is not required to be finite at t ¼ 0; since limt!0þ GðtÞ ¼ þ1; then Eq. (26) needs to be replaced by a different one. Here the method, devised in [61], is adapted to the case of Neumann boundary conditions. The three-dimensional generalization is under investigation [63]. The key steps of the Approximation Strategy can be sketched as follows: • • • • construct suitable regular approximated problems; find approximated solutions ue ; 0\e 1; show the existence of u :¼ lime!0 ue ; prove the uniqueness of the limit solution u which represents a weak solution admitted by the singular problem. Accordingly, first of all, ad hoc regular kernel problems termed approximated problems are introduced. Following the same strategy already shown in the previous Section referring to linear rigid heat conduction with memory, let Ge ðÞ :¼ Gðe þ Þ then, the integro-differential problem PeD;visco can be defined Z t PeD;visco : uett ¼ Ge ð0Þuexx þ G_ e ðt  sÞuexx ðsÞds þ f ; 0 ð31Þ together with the initial and boundary conditions ue jt¼0 ¼ u0 ðxÞ; uet jt¼0 ¼ u1 ðxÞ; uex joXð0;TÞ ¼ 0; t\T: (32) admits a unique solution. In addition, corresponding to each value of e; it is equivalent to the following integral equation Z t K e ðt  sÞuexx ðsÞds þ u1t þ u0 PeI;visco : ue ðtÞ ¼ 0 Z t Z s þ ds f ðnÞdn; ð33Þ ð32Þ The latter is a regular problem since Ge ð0Þ is finite and, therefore, the initial boundary value problem (31)– 0 where KðnÞ :¼ Z n GðsÞds Kð0Þ ¼ 0; ð34Þ 0 is well defined since G 2 L1 ð0; TÞ; 8T 2 IRþ : K is termed integrated relaxation function. Partial derivation w.r.to t, twice, of (33) delivers (31) together with initial and boundary conditions (32). Again, as observed in the case of the integral equation (25), when e ¼ 0 and the superscripts 0 are omitted, also Z t PI;visco : uðtÞ ¼ Kðt  sÞuxx ðsÞds þ u1 t þ u0 0 Z t Z s þ ds f ðnÞdn; ð35Þ 0 0 is well defined. Notably, even if the physical meaning of the involved quantities is different in the two cases, the integral equations (23) and (33) share the same terms which depend on e : the other ones, related to the different initial and boundary conditions, do not depend on e and, hence, are unchanged when the limit e ! 0 is performed. 5 Existence and uniqueness of the limit solution This Section is devoted to the existence and uniqueness of the limit solution in both the cases of onedimensional singular problems considered. Indeed, following the same method, both (23) and (33) can be proved to admit and unique solution. To improve readability, in this Section only the crucial steps to prove the results are given while details on proofs are postponed to the Appendix. Here a unified approach to both the problems is given. Accordingly, the following existence Theorem can be stated referring to both the integral problems PeI;heat as well as PeI;visco : 123 Meccanica Remark The existence and uniqueness results are proved via the same method in both the different problems (23) and (33) since they share the same integral term which depends on e; namely Z t ð36Þ K e ðt  sÞuexx ðsÞds: 0 Theorem 1 Given ue solution to the integral problem PeI;heat in (23), or PeI;visco (33), then 9 uðtÞ ¼ lim ue ðtÞ e!0 in L2 ðQÞ; Q ¼ X  ð0; TÞ: ð37Þ Proof’s Outline • • • • weak formulation, on introduction of test functions u 2 H 1 ðX  ð0; TÞÞ s.t. ux ¼ 0 on oX; consider separately the terms without e; the terms with ue and K e ; prove convergence via Lebesque’s Theorem. Furthermore, the following uniqueness Theorems, respectively, concerning the heat and viscoelastic problems can be proved. Theorem 2 The integral problems (25) as well as (35) admit a unique weak solution. Proof’s Outline in both cases, the result is proved by contradiction assuming there are two different solution and, then, showing that such an assumption leads to a contradiction. Note that the proof is unified since, on use of linearity, given two different solutions v and v~; of any of the two Eqs. (25) or (35), then their difference is again a solution. In addition, let w :¼ v  v~; it follows to satisfy Z t wðtÞ ¼ Kðt  sÞwxx ðsÞds; 0 in both the considered problems.1 6 Conclusions and perspectives A crucial role in achieving most of the results presented [59, 60], is played by the free energy. Indeed, in the case of the singular problems both in 1 Details are in the Appendix. 123 viscoelasticity and in rigid thermodynamics with memory [61, 62] are based on the free energy in one dimensional as well as in the general three-dimensional case [63]. This, aspect is one of the crucial ones which are currently under investigation, in particular, aiming to study evolution problems in the case of viscoelastic fluids as modeled in [11]. Also the a priori estimates both, in the one-dimensional as well as in higher dimensions magneto-viscoelasticity problems [59, 60], are based on the free energy. The perspective research aims to extend the study of singular kernel problems to further materials whose evolution can be modeled in a similar way. Already under investigation is the case of a magneto-viscoelasticity problem, when the singular viscoelastic behaviour is coupled with the magnetic effects [70]. Acknowledgments The partial financial support of G.N.F.M.I.N.d.A.M. and of SAPIENZA Università di Roma are gratefully acknowledged. Appendix This Appendix comprises the proofs of both Theorems in Sect. 5. They refer to the rigid heat conduction problem. Indeed, the difference is mainly in the physical meaning of the quantities involved. To prove Theorem 1, the following Lemma provides a needed estimate.2 Lemma 1 Let ue be the unique solution to the problem (18)–(19), then Z Z 1 1 jux j2 dx þ jut j2 dx ceT C ðf ; u0 Þ; ð38Þ 2 X 2 X where c ¼ maxfðkðT þ 1ÞÞ1 ; 1g: Proof The statement follows when (18) is multiplied by ut ; then, integrated over X; after use of various integrations by parts, also integrating over the time interval ð0; tÞ: Key tools are represented by the thermodynamical assumptions (20) on the heat flux relaxation function and by taking into account the assigned boundary conditions, Specifically, (18) multiplied by ut and integrated over X; when all the 2 An analogous result can be proved in the viscoelasticity problem. The Dirichlet boundary value problem with assigned initial conditions is in [61] here the Neumann problem is considered. Meccanica superscripts e are removed to simplify the notation, gives: Z Z 1d kðt þ eÞux uxt dx jut j2 dx þ 2 dt X X Z Z t _ þ eÞ½uxx ðtÞ  uxx ðt  sÞds þ ut ðtÞdx kðs X 0 Z ¼ f t ut dx; ð39Þ X which can be written as Z 1d kðt þ eÞjux j2 dx jut j dx þ 2 dt X X Z 1 2 _ þ eÞjux j dx  kðt 2 X Z t Z _ þ eÞuxt ½ux ðtÞ  ux ðt  sÞds kðs dx  0 X Z ¼ f t ut dx: 1d 2 dt Z 2 ð40Þ X In particular, the last term can be written t Z Z _ þ eÞuxt ½ux ðtÞ  ux ðt  sÞdxds kðs Z Z 1d t _ þ eÞjux ðtÞ  ux ðt  sÞj2 dx ¼ ds kðs 2 dt 0 X Z 1 _ þ eÞjux ðtÞ  ux ð0Þj2 dx þ kðt 2 X Z Z t _ þ eÞuxt ðt  sÞ½ux ðtÞ  ux ðt  sÞdxds; kðs   X 0 X 0 ð41Þ which, since which, substituted in (40) implies Z Z Z 1d 1d t 2 _ þ eÞjux ðtÞ ds kðs kðt þ eÞjux j dx  2 dt X 2 dt 0 X Z 1d ux ðt  sÞj2 dx þ jut j2 dx 2 dt X Z Z 1 _ þ eÞjux j2 dx ¼ f t ut dx þ kðt 2 X X Z Z 1 t € þ eÞjux ðtÞ  ux ðt  sÞj2 dx: ds kðs  2 0 X ð43Þ The latter, integrated over the time interval ð0; tÞ; taking into account the conditions (20), delivers Z Z 1 1 kðt þ eÞjux j2 dx þ jut j2 dx 2 X 2 X Z Z Z t 1 ð44Þ f t ut dxds þ kðeÞjuð0Þj2 dx 2 X 0 X Z 1 jf ðx; 0Þj2 dx; þ 2 X which allows to write Z Z 1 1 2 kðt þ eÞjux j dx þ jut j2 dx 2 X 2 X Z tZ  jut j2 dxds C ðf ; u0 Þ; ð45Þ X 0 and, Gronwall’s Lemma implies Z Z 1 1 kðt þ eÞjux j2 dx þ jut j2 dx 2 X 2 X eT C ðf ; u0 Þ: ð46Þ d 2uxt ðt  sÞ½ux ðtÞ  ux ðt  sÞ ¼ jux ðtÞ  ux ðt  sÞj2 ; ds Since kðt þ eÞ  kðT þ 1Þ; Z Z 1 1 jux j2 dx þ jut j2 dx ceT C ðf ; u0 Þ; 2 X 2 X can be further manipulated to deliver wherein c ¼ maxfðkðT þ 1ÞÞ1 ; 1g: Z Z t _ þ eÞuxt ½ux ðtÞ  ux ðt  sÞdxds kðs X 0 Z Z 1d t _ þ eÞjux ðtÞ  ux ðt  sÞj2 dx ds kðs ¼ 2 dt 0 X Z Z t 1 € þ eÞjux ðtÞ  ux ðt  sÞj2 dxds; þ kðs 2 X 0  ð42Þ ð47Þ h Proof of Theorem 1 The existence of the solution admitted by the singular problem is given by u ¼ limeh !0 ueh ; where ueh is a solution of (18)–(19) i.e., it solves Z t eh eh P : u ðtÞ ¼ K eh ðt  sÞuexxh ðsÞds 0 ð48Þ Z n Z t eh kðeh þ sÞds: f ðsÞds; K ðnÞ :¼ þ 0 0 123 Meccanica This thesis is proved following the outline given in Sect. 5: here each step is given. h ¼ weak formulation of the integral problem; • Z uxx Z Z Multiplication of (48) by u and integration over Q gives Z Z Z Z ueh ðtÞudxdt ¼ u Q Q Z t  Z t eh eh f ðsÞds dxdt: K ðt  sÞuxx ðsÞds þ 0 ð50Þ þ t K eh ðt  sÞueh ðsÞdsdxdt 0 Q uxx Z t ½K eh ðsÞ  KðsÞueh ðt  sÞdsdxdt 0 Q consider the test functions u; which depend on both time and space variables, subject to assigned i.c. and homogeneous Neumann boundary conditions on the boundary, oX; of X ¼ ð0; LÞ  IR; u 2 C 1 ðQÞ; Q ¼ X  ð0; TÞ; ð49Þ s:t: ux joX ¼ 0 8t 2 ð0; TÞ: 0 Z Z Z Z uxx Q Z t KðsÞueh ðt  sÞdsdxdt: 0 ð54Þ Observe that Lemma 2 Given the integral problem (48)–(49), then Z Z Z t lim uxx ½K eh ðsÞ  KðsÞueh ðt  sÞdsdxdt ¼ 0: eh !0 Q 0 ð55Þ Proof of Lemma 8ðx; tÞ 2 Q ¼ X  ð0; TÞ ¼) juj CjXj; and juxx j M; furthermore Z eh þs kðsÞds; jK eh ðsÞ  KðsÞj ¼ jK ðeh þ sÞ  KðsÞj ¼ s consider separately the terms without e; • ð56Þ The term  Z Z Z t f ðsÞds dxdt; u ð51Þ 0 Q does not depend on eh and, hence, it is unchanged in the limit eh ! 0: Furthermore, it is bounded since the history f of the material with memory is assumed to be regular and Q is bounded too. consider the term with ue and K e ; • 0 ð52Þ since it depends on e: Both the test functions u as well as ueh satisfy the homogeneous Neumann b.c. (49), then, Z Z Z t udxdt K eh ðsÞuexxh ðt  sÞds 0 Q ð53Þ Z Z Z t eh eh ¼ uxx dxdt K ðsÞu ðt  sÞds: Q 0 Now, adding and subtracting Rt eh 0 KðsÞu ðt  sÞds; the latter gives 123 Z Z Q Hence, recalling also estimate (38), the Theorem is proved that is (a) ueh ! u weakly in H 1 ð0; T; H 1 ðXÞÞ as eh ! 0; (b) ueh ! u strongly in L2 ðDÞ as eh ! 0: h the only term which needs to be considered is Z t Z t K eh ðt  sÞueh ðsÞds ¼ K eh ðsÞueh ðt  sÞds; 0 hence, since k 2 L1 ð0; TÞ; Lebesgue’s Theorem implies the limit convergence (55) and the Lemma is proved. h uxx dxdt Proof of Theorem 2 (uniqueness of the solution admitted by (25) and by (35)) The thesis is proved by contradiction; thus, assume v and v~; v 6¼ v~; both satisfy the linear equation (25). Then, also w :¼ v  v~; as any linear combination of v and v~; is a solution of the same equation. Let w :¼ v  v~; it is a solution to Z t ð57Þ wðtÞ ¼ Kðt  sÞwxx ðsÞds; 0 subject to the assigned homogeneous initial and boundary Neumann conditions. Accordingly, let X ¼ ð0; pÞ; i.e., for convenience, let L ¼ p; wx ð0; tÞ ¼ wx ðp; tÞ ¼ 0; 8t 2 ð0; TÞ: The test functions wðx; tÞ can be chosen as Meccanica wm ðx; tÞ ¼ uðtÞ cosðmxÞ; m 2 IN; ð58Þ which satisfy the assigned initial and boundary conditions, then, the solution wðx; tÞ can be written wðx; tÞ ¼ 1 X 4. an ðtÞ cosðnxÞ: ð59Þ n¼1 The weak solution, when Q ¼ ð0; pÞ  ð0; TÞ and wðx; tÞ denotes any test function, reads Z Z Z Z wðtÞwðx; tÞdxdt ¼ wðx; tÞ Q Q ð60Þ Z t Kðt  sÞwxx ðsÞdsdxdt; 0 or equivalently Z Z Z Z wðtÞwðx; tÞdxdt ¼ wxx ðx; tÞ Q Z 3. 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