Abstract A nonlinear partial differential equation of the following form is considered:

Transcription

1 M P E J Mathematical Physics Electronic Journal ISSN Volume 2, 26 Paper 5 Receive: May 3, 25, Revise: Sep, 26, Accepte: Oct 6, 26 Eitor: C.E. Wayne A Nonlinear Heat Equation with Temperature-Depenent Parameters M. A. Rincon, J. Límaco 2 & I-Shih Liu Instituto e Matemática, Universiae Feeral o Rio e Janeiro, Brazil 2 Instituto e Matemática, Universiae Feeral Fluminense, Niteroi, RJ, Brazil Abstract A nonlinear partial ifferential equation of the following form is consiere: ( ) u iv a(u) u + b(u) u 2 =, which arises from the heat conuction problems with strong temperature-epenent material parameters, such as mass ensity, specific heat an heat conuctivity. Existence, uniqueness an asymptotic behavior of initial bounary value problems uner appropriate assumptions on the material parameters are establishe for oneimensional case. Existence an asymptotic behavior for two-imensional case are also prove. Keywors: Heat equation, existence, uniqueness, asymptotic behavior. AMS Subject Classification: 35K55, 35Q8, 35B4, 35B45. Introuction Metallic materials present a complex behavior uring heat treatment processes involving phase changes. In a certain temperature range, change of temperature inuces a s: rincon@cc.ufrj.br, juanbrj@hotmail.com.br, liu@im.ufrj.br

2 phase transformation of metallic structure, which alters physical properties of the material. Inee, measurements of specific heat an conuctivity show a strong temperature epenence uring processes such as quenching of steel. Several mathematical moels, as soli mixtures an thermal-mechanical coupling, for problems of heat conuction in metallic materials have been propose, among them [6, 8, 3]. In this paper, we take a simpler approach without thermal-mechanical coupling of eformations, by consiering the nonlinear temperature epenence of thermal parameters as the sole effect ue to those complex behaviors. The above iscussion of phase transformation of metallic materials serves only as a motivation for the strong temperature-epenence of material properties. In general, thermal properties of materials o epen on the temperature, an the present formulation of heat conuction problem may be serve as a mathematical moel when the temperature-epenence of material parameters becomes important. More specifically, in contrast to the usual linear heat equation with constant coefficients, we are intereste in a nonlinear heat equation with temperature-epenent material parameters. Existence, uniqueness an asymptotic behavior of initial bounary value problems uner appropriate assumptions on the material parameters are establishe. For existence, we use particular compactness arguments for both n =, 2 cases succinctly. The tool of the proof is the compactness results of Lions, but to apply this theorem we nee a series of bouns which we establish with a number of stanar analytic techniques. For uniqueness only the case n = can be prove. The case of n = 2 remains open. The asymptotic behavior in both n =, 2 is prove employing the arguments of Lions [9] an Proi [2]. 2 A nonlinear heat equation Let θ(x, t) be the temperature fiel, then we can write the conservation of energy in the following form: ρ ε + iv q =, () where q is the heat flux, ρ the mass ensity, ε the internal energy ensity an prime enotes the time erivative. The mass ensity ρ = ρ(θ) > may epen on temperature ue to possible change of material structure, while the heat flux q is assume to be given by the Fourier law with temperature-epenent heat conuctivity, q = κ θ, κ = κ(θ) >. (2) The internal energy ensity ε = ε(θ) generally epens on the temperature, an the specific heat c is assume to be positive efine by c(θ) = ε θ 2 >, (3)

3 which is not necessarily a constant. By the assumption (3), we can reformulate the equation () in terms of the energy ε instea of the temperature θ. Rewriting Fourier law as an observing that ε = ε θ = c(θ) θ, θ we have c(θ)α(ε) = κ(θ), an hence q = κ(θ) θ = α(ε) ε, (4) α = α(ε) >. Now let u be efine as u = ε(θ), then the equation () becomes ρ(u) u iv(α(u) u) =. Since ρ(u) >, iviing the equation by ρ, an using the relation, ρ(u) iv (α(u) u) = iv ( α(u) u) ( ) ( ) α(u) u, ρ(u) ρ(u) we obtain u iv ( α(u) u) ( ) ( ) + α(u) u =. (5) ρ(u) ρ(u) Since c = ε θ = u θ, ( ρ(u) ) ρ(u) = ρ(u) 2 u u = ρ ρ(u) 2 θ Substituting into equation (5) we obtain which is equivalent to where an θ u u = ρ ρ(u) 2 θ c u. u iv ( α(u) u) ( ρ ρ(u) ρ(u) 2 θ c u) ( α(u) u ) =, u iv (a(u) u) + b(u) u 2 =, (6) a(u) = α(u) ρ(u) = α(u)c(u) ρ(u)c(u) = b(u) = α(u) ρ c ρ(u) 2 θ k(u) ρ(u)c(u) > (7) >. (8) The positiveness of a(u) an b(u) is the consequence of thermoynamic consierations (see []), an reasonable physical experiences: the specific heat c >, the thermal conuctivity κ >, the mass ensity ρ >, an the thermal expansion ρ/θ <. In this paper we shall formulate the problem base on the nonlinear heat equation (6). 3

4 2. Formulation of the Problem Let be a boune open set of IR n, n =, 2, with C bounary an Q be the cyliner (, T) of IR n+ for T >, whose lateral bounary we represent by Σ = Γ (, T). We shall consier the following non-linear problem: u iv ( a(u) u ) + b(u) u 2 = in Q, u = on Σ, (9) u(x, ) = u (x) in. Mathematical moels of semi-linear an nonlinear parabolic equations uner Dirichlet or Neumann bounary conitions have been consiere in several papers, among them, let us mention ([, 2, 3]) an ([4, 5,, 4]), respectively. Feireisl, Petzeltová an Simonon [7] prove that with non-negative initial ata, the function a(u) an g(u, u) h(u)( + u 2 ), instea of the non-linear term b(u) u 2 in (9), there exists an amissible solution positive in some interval [, T max ) an if T max < then lim u(t,.) =. t T max For Problem (9) we will prove global existence, uniqueness an asymptotic behavior for the one-imensional case (n = ) an existence an asymptotic behavior for the twoimensional case (n = 2), for small enough initial ata. 3 Existence an Uniqueness: One-imensional Case In this section we investigate the existence, uniqueness an asymptotic behavior of solutions for the case n = of Problem (9). Let ((, )), an (, ), be respectively the scalar prouct an the norms in H() an L 2 (). Thus, when we write u = u(t), u = u(t) it will mean the L 2 (), H () norm of u(x, t) respectively. To prove the existence an uniqueness of solutions for the one-imensional case, we nee the following hypotheses: H: a(u) an b(u) belongs to C (IR) an there are positive constants, a such that, a(u) a an b(u)u. H2: There is positive constant M > such that max s IR { } a u (s) ; b u (s) M. 4

5 H3: u H () H 2 (). Remark. From hypothesis (H), we have that b(u)u, u IR. Thus from hypothesis (H2), b(u) M u. Theorem Uner the hypotheses (H), (H2) an (H3), there exist a positive constant ε such that, if u satisfies ( u + u + u ) < ε then the problem (9) amits a unique solution u : Q IR, satisfying the following conitions: i. u L 2 (, T; H () H 2 ()), ii. u L 2 (, T; H ()), iii. u iv(a(u) u) + b(u) u 2 =, in L 2 (Q), iv. u() = u. Proof. To prove the theorem, we employ Galerkin metho with the Hilbertian basis from H(), given by the eigenvectors (w j ) of the spectral problem: ((w j, v)) = λ j (w j, v) for all v V = H () H2 () an j =, 2,. We represent by V m the subspace of V generate by vectors {w, w 2,..., w m }. We propose the following approximate problem: Determine u m V m, so that (u m, v) + ( a(u m ) u m, v ) + ( b(u m ) u m 2, v ) = v V m, () u m () = u m u in H () H2 (). We want to get strong convergence in H () an L 2 () of u m an u m, respectively as given later in (34). To o so, we will use the compactness results of Lions applie to a certain sequence of Galerkin approximations. First we nee to establish that they converge weakly in some particular Sobolev spaces, an we o this through energy estimates. Existence The system of orinary ifferential equations () has a local solution u m = u m (x, t) in the interval (, T m ). The estimates that follow permit to exten the solution u m (x, t) to interval [, T[ for all T > an to take the limit in (). Estimate I: Taking v = u m (t) in equation () an integrating over (, T), we obtain T T 2 u m 2 + u m 2 + b(u m )u m u m 2 < 2 u 2, () where we have use hypothesis (H). Taking â = min{, } >, we obtain 2 u m 2 + T u m 2 + T b(u m )u m u m 2 < 2â u 2. (2) 5

6 Thus, applying the Gronwall s inequality in (2) yiels (u m ) is boune in L (, T; L 2 ()) L 2 (, T; H ()). (3) Estimate II: Taking v = u m in equation () an integrating over, we obtain u m 2 + a(u m ) u m 2 = a 2 t 2 u (u m)u m u m 2 b(u m )u m u m 2. (4) On the other han, from hypothesis (H2), we have the following inequality, a 2 u (u m)u m u m 2 2 MC u m u m 2, (5) an since L () C an b(u m ) M u m, we obtain b(u m )u m u m 2 MC 2 u m u m 3, (6) where C = C () is a constant epening on. Substituting (5) an (6) into the right han sie of (4) we get u m t a(u m ) u m 2 2 MC u m u m 2 + MC 2 u m u m 3 4 (MC ) 2 u m 2 u m u m 2 + (MC 2 )2 u m 2 u m 4. (7) Now taking the erivative of the equation () with respect to t an making v = u m, we have 2 t u m 2 + a(u m ) u m 2 b = u (u m) u m 2 u m 2 a u (u m) u m u m 2 2 b(u m ) u m u m u (8) m MC 2 (3 + M 2 ) u m 2 u m u m 2. From the inequality (7) an (8), we have { t 2 u m 2 + } a(u m ) u m 2 2 u m 2 { a 2 α u m 4 α u m 2 + u m u m 2 + } u m 2, (9) where we have efine α = (MC 2 ) 2 an α = MC ( 3 4 MC + 3C ). 6

7 Now, uner the conition that the following inequality, α u m 4 + α u m 2 <, t, (2) 4 be vali, the coefficients of the term u m in the relation (9) is positive an we can integrate it with respect to t, u m 2 + t a(u m ) u m u m 2 + 3a t u 2 m 2 C. (2) Therefore, applying the Gronwall s inequality in (2), we obtain the following estimate: (u m ) is boune in L (, T; L 2 ()) L 2 (, T; H ()). (22) Now we want to prove that the inequality (2) is vali if the initial ata are sufficiently small. In other wors, there is some ε > such that ( u + u + u ) < ε an the following conition hols, an { α S a 2 + a u 2 + } 2 u 2 + α { S + a u 2 + u 2} < 4 (23) α u 4 + α u 2 < 4, (24) where we have enote S = ( M( u + u 2 + u u 2 ) + a u ) 2. We shall prove this by contraiction. Suppose that (2) is false, then there is a t such that α u m (t) 4 + α u m (t) 2 < if < t < t (25) 4 an α u m (t ) 4 + α u m (t ) 2 = 4. (26) Integrating (9) from to t, we obtain 2 u m (t ) t a(u m ) u m (t ) 2 2 u m () 2 + a(u m ()) u m () 2 u m 2 ( M( u + u 2 + u u 2 ) + a u 2 ) 2 +a u 2 + u 2, (27) an consequently, u m (t ) 2 S + a u 2 + u a 2 2. (28) 7

8 Using (23) an (24) we obtain α u m (t ) 4 +α u m (t ) 2 α a 2 +α { S + a u 2 + u 2 hence, comparing with (26), we have a contraiction. {S + a u 2 + a u 2 } 2 } < 4, Estimate III: Taking v = u m (t) in the equation (), an using hypothesis H, we obtain t u m 2 + a(u m ) u m 2 a u m u m + b u m 2 u m, where we have use the following inequality, b(u m ) u m 2 u m ˆb u m L u m u m ˆb u m H () u m u m ˆb u m u m u m = ˆb u m u m 2. In this expression we have enote ˆb = sup b(s), for all s [, ]. Note that, 4α 4α from (2), we have u m (t) u m (t) < /4α an b = ˆb + M. Using hypothesis (H), we obtain or equivalently, t u m u m 2 a u m u m + b u m 2 u m, ( ) t u m 2 a + 2 b u m u m 2 a u m u m. Similar to the choice of ε for the conitions (23) an (24), ε will be further restricte to guarantee that the following conition also hols, b ( S + a u 2 + u 2) /2 < 2 4 an from (2) we obtain a ( ) 4 a 2 b u m, it implies that (29) (3) t u m u m 2 a u m u m 8 u m 2 + C u m 2. (3) Now, integrating from to t, we obtain the estimate u m 2 + T u m 2 Ĉ. 8

9 Hence, we have (u m ) is boune in L (, T; H ()), (u m ) is boune in L 2 (, T; H () H2 ()). (32) Limit of the approximate solutions From the estimates (3), (22) an (32), we can take the limit of the nonlinear system (). In fact, there exists a subsequence of (u m ) m N, which we enote as the original sequence, such that u m u weak star in L (, T; L 2 ()), u m u weak in L 2 (, T; H ()), u m u weak star in L (, T; H ()), u m u weak in L 2 (, T; H () H2 ()). (33) Thus, by compact injection of H ( (, T)) into L 2 ( (, T)) it follows by compactness arguments of Aubin-Lions [9], we can extract a subsequence of (u m ) m N, still represente by (u m ) m N such that u m u strong in L 2 (, T; H ()) an a.e. in Q, u m u strong in L 2 (Q) an a.e. in Q. (34) Let us analyze the nonlinear terms from the approximate system (). From the first term, we know that a(u m ) u m 2 a 2 u m 2 a 2 C, (35) an since u m u a.e. in Q an a(x,.) is continuous, we get a(u m ) a(u) an u m u a.e. in Q. (36) Hence, we also have a(u m ) u m 2 a(u) u 2 a.e. in Q. (37) From (35) an (37), an Lions Lemma, we obtain a(u m ) u m a(u) u weak in L 2 (Q). (38) From the secon term, we know that T b(u m ) u m 2 2 b 2 T u m 2 L () u m 2 T T b 2 C u m 2 u m 2 H b2 C u m 2 u m 2 C, (39) 9

10 where C has been efine in (6). By the same argument that leas to (37), we get b(u m ) u m 2 2 b(u) u 2 2 a.e. in Q (4) Hence, from (39) an (4) we obtain the convergence, b(u m ) u m 2 b(u) u 2 weak in L 2 (Q). (4) Taking into account (33), (38) an (4) into (), there exists a function u(x, t) efine over [, T[ with value in IR satisfying (9). Moreover, from the convergence results obtaine, we have that u m () = u m u in H () H 2 () an the initial conition is well efine. Hence, we conclue that equation (9) hols in the sense of L 2 (, T; L 2 ()). Uniqueness Let w(x, t) = u(x, t) v(x, t), where u(x, t) an v(x, t) are solutions of Problem (9). Then we have w iv (a(u) w) iv(a(u) a(v)) v + b(u) ( u 2 v 2 ) + (b(u) b(v)) v 2 = in Q, w = on Σ, w(x, ) = in. Multiplying by w(t), integrating over, we obtain 2 t w 2 + a(u) w 2 a (û) w v w u + b(u) ( u + v ) w w + v 2 b (u) w 2 u ( ) M w w v + C u + v w w + M M v L () w w + C ( u L () + v L ()) w w v 2 w 2 (42) (43) +M v 2 L () w 2 2 w 2 + C( u 2 + v 2 ) w 2, where we have use (a) The generalize mean-value theorem, i.e., a(u) a(v) = a (û)(u v) a(û) w, u û v, u u (b) b(u) ( u 2 v 2 ) b(u) w ( u + v ),

11 (c) v L () v H () v H 2 () v L 2 (). The last inequality is vali only for one-imensional case. Integrating (43) from to t, we obtain 2 w 2 + a t w 2 t 2 2 w() 2 + C ( u 2 + v 2 ) w 2, where C enotes a ifferent positive constant. Since w() = u v =, using the Gronwall s inequality, we obtain w 2 + t w 2 =, which implies the uniqueness, w(x, t) = u(x, t) v(x, t) = an the theorem is prove. Asymptotic behavior In the following we shall prove that the solution u(x, t) of Problem (9) ecays exponentially when time t, using the same proceure evelope in Lions [9] an Proi [2]. Thus, we will initially show the exponential ecay of the energy associate with the approximate solutions u m (x, t) of Problema (). Theorem 2 Let u(x, t) be the solution of Problem (9). Then there exist positive constants ŝ an Ĉ = Ĉ{ u, u } such that u 2 + u 2 Ĉ exp ŝ t. (44) Proof. To prove the theorem, complementary estimates are neee. Estimate I : Consier the approximate system (). Using the same argument as Estimate I, i.e, taking v = u m (t), we have t u m 2 + a(u m ) u m 2 + b(u m )u m u m 2 =. (45) Integrating (46) from (, T), we obtain T T 2 u m 2 + u m 2 + b(u m )u m u m 2 < 2 u 2. (46) From H hypothesis, (45) an (46) we conclue u m 2 2 u m u m 2 u m u. (47)

12 Estimate II : Taking erivative of the system () with respect to t an making v = u m, we obtain t u m 2 + u m 2 M ( u m 2 u m + u m 2 u m 2) ( (48) +2M u m u m u m C u m 2 u m + u m 2 u m 2), where C = C (M, ). Hence, t u m 2 + a ( ) 2 u m 2 + u m 2 a 2 C u m C u m 2. (49) Using (47) then we can write the inequality (49) in the form, t u m u m 2 + u a m 2( 2 C ( 2 u ) /2 u a m /2 2 u C u m a ) (5) Let v = u m() in (). Then where C = C(a, M, ), an We efine the operator u m () 2 C ( u + u + u 2 + u 3) u m (), u m () 2 ( C ( u + u + u 2 + u 3) ) 2. (5) J(u ) = Then we have shown that ( ) /2 2 u ( C ( u + u + u 2 + u 3 ) ) /2 + 2 u C ( u + u + u 2 + u 3). (52) ( 2 u ) /2 u m () /2 + 2 u u m () J(u ). (53) If we choose C a positive constant an u small enough, so that the following inequality hols, C J(u ) < 4, (54) ( ) /2 2 u C u 2 u m a /2 + C u m a <, t. (55) 4 2

13 Inee, we can prove this by contraiction. Suppose that there is a t such that C ( 2 u ) /2 u m (t ) /2 + C 2 u u m(t ) = 4. (56) Integrating (5) from to t, we obtain u (t ) 2 u () 2. From (54) an (55), we conclue that C ( 2 u ) /2 u m (t ) /2 + C 2 u u m (t ) C ( 2 u ) /2 u m () /2 + C 2 u u m() C J(u ) < 4. (57) Therefore, we have a contraiction by (56). From (5), (55) an using the Poincaré inequality, we obtain t u m 2 + s u m 2 (58) where s = ( c )/2 an c is a positive constant such that H () c L 2 (). Consequently, we have { exp s t u m t 2} (59) an hence u m 2 u m() 2 exp s t ( C ( u + u + u 2 + u 3 ) ) 2 exp s t C ( u, u ) exp s t, (6) where we have use the inequality (5). We also have from (47) that u m 2 2 u u m 2 C ( u, u ) u exp s t/2. (6) Defining ŝ = s /2 an Ĉ = C + 2 C then the result follows from (6), (6) inequality an of the Banach-Steinhaus theorem. 4 Existence: Two-imensional Case In this section we investigate the existence an asymptotic behavior of solutions for the case n = 2 of Problem (9). In orer to prove these results we nee the following hypotheses: 3

14 H: Let a(u) belongs to C 2 [, ) an b(u) belongs to C [, ) an there are positive constants, a such that, a(u) a an b(u)u. H2: There is a positive constant M > such that H3: a () =. u H4: u H() H 3 (). max s IR { a ; u (s) b ; u (s) 2 } a u 2(s) M. Theorem 3 Uner the hypotheses (H) - (H4), there exists a positive constant ε such that, if u satisfies ( u + u H 3 ()) < ε, then Problem (9) amits a solution u : Q IR, satisfying the following conitions: i. u L (, T; H () H2 ()), ii. u L 2 (, T; H () H 2 ()), iii. u iv(a(u) u) + b(u) u 2 = in L 2 (Q), iv. u() = u. Proof. To prove the theorem, we employ Galerkin metho with the Hilbertian basis from H (), given by the eigenvectors (w j) of the spectral problem: ((w j, v)) = λ j (w j, v), for all v V = H () H2 ()) an j =, 2,. We represent by V m the subspace of V generate by vectors {w, w 2,..., w m }. Let u m (x, t) be the local solution of the approximate problem (). With similar arguments for the one-imensional case, in orer to exten the local solution to the interval (, T) inepenent of m, the following a priori estimates are neee. Estimate I: Taking v = u m (t) in the equation () an integrating over (, T), we obtain T T 2 u m 2 + u m 2 + b(u m )u m u m 2 < 2 u 2. (62) Using the hypothesis (H), we have (u m ) is boune in L (, T; L 2 ()) L 2 (, T; H ()). (63) Estimate II: Taking v = u m (t) in (), we obtain 2 t u m 2 + a(u m ) u m 2 = b(u m ) u m 2 a u m + u u m 2 u m (64) 4

15 From hypothesis (H) an using Sobolev embeing theorem we have the inequality b(u m ) u m 2 u m M u m u m 2 u m M u m L 6 () u m L 6 () u m L 6 () u m L 2 () MC 3 u m u m 2 H 2 () u m C u m 4, where C = MC 3 C 2 C 3 >, since C, C 2, C 3 are positive constants satisfying the following inequalities: (65) u m L 6 C u m H, u m H 2 C 2 u m an u m H C 3 u m. (66) We also have that a u u m 2 u m M u m 2 u m M u m L 4 () u m L 4 () u m MC 2 4 u m H 2 () u m H 2 () u m MC 2 4C 2 2 u m 3 (67) C 5 u m 3 C2 5 u m u m 2. where C 5 = MC4C > an C 4 is a positive constant satisfying the inequality: u m L 4 () C 4 u m H (). Substituting (65) an (67) in the equality (64) an using the hypothesis (H), we obtain 2 t u m u m 2 ( C + C2 ) 5 um 4. (68) Consier now v = u m in (). Integrating in, we obtain u m 2 + a(u m ) u m 2 = b(u m ) u m 2 u m 2 t + a u u m 2 u m + 2 a u u m u m 2. In the following, we shall get estimates for the first, the secon an the thir terms on the han right sie of (69). For the first term, from hypothesis (H2) an using Sobolev embeing theorem, we obtain b(u m ) u m 2 u m M u m u m 2 u m (69) M u m L 6 () u m L 6 () u m L 6 () u m L 2 () MC 3 u m u m 2 H 2 () u m C u m 3 u m (7) C2 2 u m u m 2 u m 2, 5

16 where C = MC 3 C 2 C 3 >, was efine in (65). For the secon term on the han right sie of (69), we have a u u m 2 u m M u m 2 u m M u m 2 L 4 () u m L 2 () MC 2 4 u m 2 H 2 () u m L 2 () C 5 u m 2 u m (7) 2C2 5 u m u m 2, where C 4 an C 5 are positive constants efine in (66) an (67). For the thir term on the han right sie of (69), we have 2 a u u m u m 2 M 2 u m u m 2 M 2 u m C () u m 2 M 2 C 6 u m H 2 () u m 2 M 2 C 6C 2 u m u m 2 (72) C 7 u m u m 2 2C2 7 u m 4 + 8, u m 2, where C 6 is the positive constant of embeing between spaces, H 2 () C (), i.e, u C () C 6 u H 2 () an C 7 = M 2 C 6C 2 >. Substituting (7), (7) an (72) in (69), we obtain u m t a(u m ) u m 2 ( C C C 2 7 ) um u m u m 2 u m 2. On the other han, taking erivative in the equation () with respect to t an taking v = u m an integrating in, we have 2 t u m 2 + a(u m ) u a m 2 = u u m u m u m 2 a u 2 u m u m 2 u m 2 a u u m u m u m b u u m u m 2 u m 2 b(u m ) u m u m u m. Again, we shall estimate the five terms on the han right sie of (74) iniviually. For (73) (74) 6

17 the first term, from hypothesis (H2) an using Sobolev embeing theorem we obtain a u u m u m u m M u m C () u m u m MC 6C 2 u m u m 2 For the secon term, we have 2 a u 2 u m u m 2 u m 2M2 C 2 6 C2 2 u m 2 u m u m 2. M u m L 6 () u m 2 L 6 () u m L 2 () MC 2 2 C C 3 u m 2 u m 2. For the thir term, a u u m u m u m M u m L 4 () u m 2 L () u m 4 L 2 () For the fourth term, MC 2 4C 2 2 u m u m 2 6 u m 2 + 4M2 C 4 4C 4 2 u m 2 u m 2. (75) (76) (77) b u u m u m 2 u m MC2 2 C C 3 u m 2 u m 2. (78) For the last term of (74), we have b(u m ) u m u m u m MC4 2C2 2 u m u m 2 Substituting the estimates (75)-(79) in (74), we obtain 2 t u m 2 + where C 8 = 2 ( M 2 C 2 6 C2 2 + MC 2 2C C 3 + 8M2 C 4 4 C4 2 ). 6 u m 2 + 4M2 C 4 4 C4 2 u m 2 u m 2. (79) a(u m ) u m 2 C 8 u m 2 u m u m 2, (8) Aing the estimates (68), (73) an (8), we obtain ( um 2 + u m 2 t 2 + a(u m ) u m 2) u m u m 2 C 9 u m 4 + C 8 u m 2 u m 2, (8) where C 9 = ( 3C C2 7 + C2 2 + C ). 7

18 From estimate (8), we obtain ( um 2 + u m 2 t 2 + a(u m ) u m 2) + 2 u m 2 + u m 2( 4 C 9 u m 2) + 8 u m 2 + u m 2( 4 C 8 u m 2). On the other han, making t = in the equation (), taking v = u m (), integrating in an using the hypothesis (H3), we have u m () 2 = a (u m ) u m u m () + b (u m ) u m 2 u m () = a + a (u m ) ( u m ) u m () a u (u m) ( u m 2 ) u m() b(u m ) ( u m 2) u m(). a u (u m)( u m ) 2 u m () u (u m) u m u m () u m () 2 a u 2(u m u m 2 u m u m () b u (u m) u m 2 u m u m() From (83), we obtain the following estimate for the u m() term, Or equivalently u m () 2 C ( um H 3 () u m + u m H 3 () + u m 3) u m (). (82) (83) u m() 2 C 2 ( um H 3 () u m + u m H 3 () + u m 3) 2, (84) where C = max { MC 2 4 C 2 + 2MC 4 C 2 + MC 2 C 3C 2, M, 2MC C 2 }. There is ε > such that for ( u + u H 3 ()) < ɛ, we have u 2 < (C 8 + C 9 ) 4, ( u + C 2 u H 3 () u + u H 3 () + u 3) 2 + a u 2 < (C 8 + C 9 ) 4. (85) Therefore, we can confirm that u m 2 < (C 8 + C 9 ), t. (86) 4 8

19 Inee, by presuming absurity, there is a t by (86) such that u m (t) 2 < (C 8 + C 9 ) 4, if < t < t an (87) u m (t ) 2 = (C 8 + C 9 ) 4. Then, by integrating (82) from to t an using hypothesis (H), we obtain u m (t ) 2 + u m (t ) 2 + u m (t ) 2 u m 2 + u m 2 + a u m 2. (88) From (84), (85) an (88) we obtain ( um (t ) 2 + u m 2 (t ) 2 + u m (t ) 2) < 2 (C 8 + C 9 ) 4. (89) Therefore, we conclue that u m (t ) 2 < (C 8 + C 9 ) 4. This leas to a contraiction by (87) 2. Since (86) is vali, the terms ( 4 C 8 u m 2) an ( 4 C 9 u m 2) on the left han-sie of (82) are also positive. Hence, by integrating (82) from to T, we obtain Therefore, 2 u m 2 + u m u m 2 + a T u m 2 + a T u m C. (9) (u m ) is boune in L (, T; H () H 2 ()), (u m ) is boune in L2 (, T; H () H2 ()) L (, T; H ()). (9) The limit of the approximate solutions can be obtaine following the same arguments for (33), (34), (38) an (4), i.e, we obtain the solution in L 2 (Q). Asymptotic behavior In the following we shall prove that the solution u(x, t) of Problem (9), in the case n = 2, also ecays exponentially when time t. Theorem 4 Let u(x, t) be the solution of Problem (9). Then there exist positive constants S an C = C{ u, u, u H 3 ()} such that u 2 + u 2 + a(u) u 2 C exp S t. (92) 9

20 Proof. Let H(t) = 2( um 2 + u m 2 + From (82), we obtain a(u m ) u m 2). We also have t H(t) + u m u m 2. (93) H(t) ( 2 C2 3 um 2 + u m 2) + a 2 u m 2 Ĉ( um 2 + u m 2) ( Ĉ2 2 u m u m 2), (94) where Ĉ = max { C3, 2 a } an Ĉ 2 = 8Ĉ. Using (93) an (94), we conclue that 2 which implies that 2 t H(t) + Ĉ2H(t), H(t) C exp S t, where S = 2Ĉ2 an the positive constant C = C{ u, u, u H 3 ()} is etermine by (84). The result follows from the Banach-Steinhaus theorem. Acknowlegement: We woul like to thank the reviewer who carefully went through our paper an gave valuable comments an suggestions. References [] Alaa, N., Iguername, M., Weak perioic solution of some quasilinear parabolic equations with ata measures, Journal of Inequalities in Pure an Applie Mathematics, 3, (22), -4. [2] Amann, H. Perioic solutions of semilinear parabolic equations, Nonlinear Analysis, (978), -29. [3] Boccaro, L., Murat, F., Puel, J.P. Existence results for some quasilinear parabolic equations, Nonlinear Analysis, 3, (989), [4] Boccaro, L., Gallouet, T. Nonlinear elliptic an parabolic equations involving measure ata, Journal of Functional Analysis, 87, (989), [5] Chipot,M., Weissler, F.B. Some blow up results for a nonlinear parabolic equation with a graient term, SIAM J. Math. Anal., 2, (989),

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