Electronic Journal of Qualitative Theory of Differential Equations

In this paper we investigate the existence and uniqueness of global solutions, and a rate stability for the energy related with a Cauchy problem to the viscous Burgers equation in unbounded domain R × (0, ∞). Some aspects associated with a Cauchy problem are presented in order to employ the approximations of Faedo-Galerkin in whole real line R. This becomes possible due to the introduction of weight Sobolev spaces which allow us to use arguments of compactness in the Sobolev spaces.


Introduction and Formulation of the Problem
We are concerned with the existence of global solutions -precisely, global weak solutions, global strong solutions and regularity of the strong solutions -, uniqueness of the solutions and the asymptotic stability of the energy for the nonlinear Cauchy problem related to the classic viscous Burgers equation u t + uu x − u xx = 0 established in R × (0, T ), for an arbitrary T > 0.More precisely, we consider the real valued function u = u(x, t) defined for all (x, t) ∈ R × (0, T ) which is the solution of the Cauchy problem u t + uu x − u xx = 0 in R × (0, T ), u(x, 0) = u 0 (x) in R. (1.1) 1 Universidade Federal Fluminense, IM, RJ, Brasil 2 Corresponding author: hclark@vm.uff.br 3 Universidade Federal do Rio de Janeiro, IM, RJ, Brasil, lmedeiros@abc.org.brEJQTDE, 2010 No. 18, p. 1 The Burgers equation has a long history.We briefly sketch this history by citing one of the pioneer work by Bateman [2] about an approximation of the flux of fluids.
Later, Burgers published the works [5] and [6] which are also about flux of fluids or turbulence.In the classic fashion the Burgers equation has been studied by several authors, mainly in the last century, and excellent papers and books can be found in the literature specialized in PDE.One can cite, for instance, Courant & Friedrichs [8], Courant & Hilbert [9], Hopf [10], Lax [13] and Stoker [17].
Today, the equation (1.1) 1 ((1.1) 1 refers to the first equation in (1.1)) is known as viscous Burgers equations and perhaps it is the simplest nonlinear equation associating the nonlinear propagation of waves with the effect of the heat conduction.
The existence of global solutions for the Cauchy problem (1.1) will be obtained employing the Faedo-Galerkin and Compactness methods.The Faedo-Galerkin method is probably one of the most effective methods to establish existence of solutions for nonlinear evolution problems in domains whose spatial variable x lives in bounded sets.To spatial unbounded sets, there exist few results about existence of solutions established by the referred method.Thus, as the non-linear problem (1.1) is defined in R, in order to reach our goal through this method we will also need to use compactness' argument, as in Aubin [1] or Lions [15].In order to apply the Compactness method we employ a suitable theory on weight Sobolev spaces to be set as follows.In fact, in the sequel is the Lebesgue space of the classes of functions u : R → R with square integrable on R. Assuming that X is a Banach space, T is a positive real number or T = +∞ and 1 ≤ p ≤ ∞, we will denote by L p (0, T ; X) the Banach space of all measurable mapping u :]0, T [−→ X, such that t → u(t) X belongs to L p (0, T ).For more details on the functional spaces above cited the reader can consult, for instance, the references [3] and [15].In this work we will also use the following weight vectorial spaces where K is a weight function given for The inner product and norm of L 2 (K) and H m (K) are defined by respectively.The vector spaces L 2 (K) and H m (K) are Hilbert spaces with the above inner products.By D(R) it denotes the class of C ∞ functions in R with compact support and convergence in the Laurent Schwartz sense, see [16].
We will also use the functional structure of the spaces L p (0, T ; where H is one of the spaces: Some properties of the spaces L 2 (K) and H m (K) as the compactness of the inclusion H m (K) ֒→ L 2 (K) and Poincaré inequality with the weight (1.2) has been proven in Escobedo-Kavian [11].Results on compactness of space of spherically symmetric functions that vanishes at infinity were proven by Strauss [18].In this direction one can see some results in Kurtz [12].
The method used to prove the existence of solutions for the Cauchy problem (1.1) is to transform it to another equivalent one proposed in the suitable functional spaces by using a change of variables defined by z(y, s) = (t + 1) 1/2 u(x, t) where y = x (t + 1) 1/2 and s = ln(t + 1). (1. 3) The changing of variable (1.3) defines a diffeomorphism σ : R x × (0, T ) → R y × (0, S) with σ(x, t) = (y, s) and S = ln(T + 1).From (1.3) we have t = e s − 1 and x = e s/2 y.
Differentiating u with respect to t, it yields Differentiating again with respect to x, it yields ∂y ∂x = (t + 1) −3/2 z yy = e −3s/2 z yy .
Inserting the three last identities in (1.1) 1 , we obtain Moreover, for t = 0, we have by definition of y that x = y.Thus the initial data becomes u 0 (x) = u(x, 0) = z(y, 0) = z 0 (y). (1.5) For use later and a better understanding we will modify the equation (1.4) as follows: one defines the operator L : , which satisfies: for all φ ∈ H 2 (K) and ψ ∈ H 1 (K).Therefore, from (1.4), (1.5) and (1.6) 1 the Cauchy The purpose of this work is: in Section 2, we investigate the existence of global weak solutions of (1.1), its uniqueness and as well as analysis of the decay of these solutions.
In Section 3 we establish the same properties of Section 2 for the strong solutions.In Section 4, we study the regularity of the strong solutions.

Weak Solution
Setting the initial data u 0 ∈ L the function u satisfies the identity integral for all v ∈ H 1 (R) and for all ϕ ∈ D(0, T ).Moreover, u satisfies the initial condition The existence of solution of (1.1) in the precedent sense is guaranteed by the following theorem Theorem 2.1.Suppose u 0 ∈ L 2 (K), then there exists a unique global solution u of (1.1) in the sense of Definition 2.1.Moreover, energy (2. 2) The following proposition, whose proof has have been done in Escobedo & Kavian [11], will be useful throughout this paper.
Proposition 2.1.One has the results The eigenvalues of L are positive real numbers λ j = j/2 for j = 1, 2 . . ., and the related space with As the two Cauchy problems (1.
the function z satisfies the identity integral for all v ∈ H 1 (K) and for all ϕ ∈ D(0, S).Moreover, z satisfies the initial condition z(y, 0) = z 0 (y) for all y ∈ R.
The existence of solutions for system (1.7) will be shown by means of Faedo-Galerkin method.In fact, as L 2 (K) is a separable Hilbert space there exists a orthogonal hilber- there exist ω j solutions of the spectral problem associated with the operator L in H 1 (K).
This means that (Lω j , v) = λ j (ω j , v) for all v ∈ H 1 (K) and j ∈ N. (2.4) In these conditions one defines V N as the subspace of L 2 (K) spanned by the N − eigenfunction ω 1 , ω 2 , . . ., ω N of (ω j ) j∈N , being ω j with j ∈ N defined by (2.4).Now, we are ready to state the following result. where for some C 1 > 0 established to follow at the end of the proof below.
then from Proposition 2.1 one has As the continuous immersion . This proves the statement (a).As K 1/2 ≥ 1, then from (a) one gets (b).The statement (c) is an immediate consequence from Proposition 2.1-item (5).Notice that for all v ∈ H 2 (K) one has (v y , v y ) = (Lv, v) .From this and Proposition 2.1-item (5) one gets (d).Finally, let v ∈ H 2 (K) and Lv = f with f ∈ L 2 (K).Defining w = K 1/2 v one can write From this, Proposition 2.1-item (5) and Proposition 2.2-item (c), one has On the other hand, one has From these two above inequalities, Proposition 2.1-item ( 5) and (d) one obtains (e) EJQTDE, 2010 No. 18, p. 7 Proof of Theorem 2.2 -We will employ the Faedo-Galerking approximate method to prove the existence of solutions.In fact, the approximate system is obtained from (2.4) and this consists in finding z N (s, y) = N i=1 g iN (s)ω i (y) ∈ V N , the solution of the system of ordinary differential equations for all ω belong to V N .The System (2.6) has local solution z N in 0 ≤ s < s N , see for instance, Coddington-Levinson [7].The estimates to be proven later allow us to extend the solutions z N to whole interval [0, S[ for all S > 0 and to obtain subsequences that converge, in convenient spaces, to the solution of (1.7) in the sense of Definition 2.2.
The integral above is upper bounded.In fact, by using Hölder inequality, Proposition 2.1-item (5) and Proposition 2.2-item (b) we can write From this and from precedent identity we get 1 2 By using (1.6) 2 , the fact that basis (ω j ) is orthonormal and (2.4) we have By using these two identities we are able to prove that In fact, note that (g jN (s)) 2 (λ j − 1) ≥ 0 for all s and N. (2.9) From Proposition 2.1-item (4) the second statement in (2.9) is obvious.Therefore, it suffices to prove that g 1N (s) = 0 for all s and N.In fact, first, note that Thus, we will show that (z N (s), (2.10) By using (2.4) and Proposition 2.1-item (4) one can writes The non-linear term of (2.10) is null, because we have used above Proposition 2.1-item (4), that is, From this, as ω j ∈ H 1 (K) then z N 2 and z N y 2 belong to L 1 (R) and consequently lim |y|−→∞ z N (y, t) = 0. Thus, z N (s)z N y (s), ω 1 = 0 for all N and s.Taking into account these facts in (2.10), it yields z N s (s), ω 1 = 0. Thus, by using (2.6) 2 and hypothesis on z 0 we get z N (s), ω 1 = z N (0), ω 1 = 0. Therefore, this completes the proof of statement of (2.9) Since (2.8) is true, the inequality (2.7) is reduced to Next, we will prove that for all s ≥ 0.
(2.12) EJQTDE, 2010 No. 18, p. 9 In fact, suppose it is not true.Then there exists s * such that Integrating (2.11) from 0 to s * , it yields From hypothesis on z 0 we have This contradicts z N (s * ) = 1/4 √ 3C 1 .Thus, (2.12) it is true.Therefore, integrating (2.11) from 0 to s and by using (2.4) and (2.6) 2 , it yields Estimate 2. In this estimate we will use the projection operator Thus, from (2.6) 1 we have From this and definition of P N one can write The identity (2.14) is verified in the L ∞ 0, S; H 1 (K) ′ sense.In fact, analyzing each term on the right-hand side of (2.14) we prove this statement as one can see: (2.16) (2.17) As the proof of the three identities (2.15), (2.16) and (2.17) are similar, we will just make the last one.In fact, On the other hand, EJQTDE, 2010 No. 18, p. 11 Inserting this inequality in the precedent one we get (2.17).By using (2.15)-(2.17) in (2.14), we get where we have used in the two last step the Poincaré inequality and Estimate (2.13).
Integrating this inequality from 0 to S and again using Estimate (2.13), we obtain where The limit in the approximate problem (2.6):By Estimates 1 and 2, more precisely, from (2.13) and (2.18) we can extract subsequences of (z N ), which one will denote by (z N ), and a function z : R × (0, S) → R satisfying (2.19) From these convergence we are able to pass to the limits in the linear terms of (2.6).The nonlinear term needs careful analysis.In fact, from (2.19) 1, 3 and Aubin's compactness result, see Aubin [1], Browder [4], Lions [15] or Lions [14], we can extract a subsequences of (z N ), which one will denote by (z N ), such that

.20)
On the other hand, for all φ(x, s) = v(x)θ(s) with v ∈ H 1 (K) and θ ∈ D(0, S) we have EJQTDE, 2010 No. 18, p. 12 Next, we will show that the last two integrals on the right-hand side of (2.21) converge.
In fact, the first one can be upper bounded as follows From this, (2.13) and (2.20) we have The second integral also converges because from (2.19) 2 we have, in particular, that and because φz ∈ L 2 0, S; L 2 (K) .Therefore, we have Taking the duality paring on the both sides of (2.22) 1 with ϕ we obtain 1 2 Substituting this inequality in (2.23) yields 1 2 From (2.19) 2 one has that z and z belong to L 2 0, S; H 1 (K) .Therefore, applying the Gronwall inequality one gets ϕ(s) = 0 in [0, S] Asymptotic behavior: The asymptotic behavior, as s → ∞, of E(s) = 1 2 |z(s)| 2 given by the unique solution of the Cauchy problem (1.7) is established as a consequence of inequality (2.11).In fact, from (2.11), (2.12) and Banach-Steinhauss theorem we get that the limit function z defined by (2.19)

Strong Solution
Setting the initial data u 0 ∈ H 1 (K) we are able to show that the Cauchy problem (1.1) has a unique real valued strong solution u = u(x, t) defined in R × (0, T ) for all T > 0. Precisely, the strong solution of (1.1) is defined as follows.
Definition 3.1.A global strong solution for the initial value problem (1.1) is a real valued function u = u(x, t) defined in R × (0, T ) for an arbitrary T > 0, such that for all ϕ ∈ L 2 0, T ; H 1 (R) .Moreover, u satisfies the initial condition u(x, 0) = u 0 (x) for all x ∈ R.
The existence of solution of (1.1) in the precedent sense is guaranteed by the following theorem.
EJQTDE, 2010 No. 18, p. 15 The existence of solutions of the system (1.7) will be also shown by means of Faedo-Galerkin method and by using the special basis defined as solutions of spectral problem (2.4) and the first eigenfunction Under these conditions one defines V N as in Section 2. Now we state the following theorem.
Theorem 3.2.Suppose z 0 ∈ H Proof of Theorem 3.2 -We need to establish two estimates.In fact, Next, we will find the upper bound of the last term on the right-hand side of the above inequality.In fact, by using Hölder inequality, Proposition 2.2-items (b), (c) and (d) From this we have 1 2 Use a similar argument as in Estimate 1 we are able to prove that In fact, note that From (2.9) we have N j=2 (g jN (s)) 2 λ 2 j − 1 ≥ 0 and g 1N (s) = 0 for all s and N.
From this we obtain (3.4), see Estimate 1.Since (3.4) is true, the inequality (3.3) is reduced to Next, proceeding as (2.12) we will prove that Next, we will estimate the three inner product on the right-hand side of the above identity.In fact, by usual inequalities and Proposition 2.2 we have , where C 2 = max{1, √ 3C 1 }.From this we get a constant C > 0 independent of N and s such that  From (3.7) and (3.8) we can take the limit on the approximate system (2.6).In fact, the analysis of the limit as N −→ ∞ in the linear terms of (2.6) is similar to those of Section 2. However, the nonlinear term is made as follows.From (3.7), (3.8) and Aubin-Lions theorem one extracts subsequences of (z N ), which will be denoted by

Estimate 4 .
s)| 2 ≤ 0.Integrating from 0 to s and using the hypothesis on the initial data we obtain Settingω = z N s (s) ∈ V N in (2.6) 1 , it yields |z N s (s)| 2 = − Lz N (s), z N s (s) + 1 2 z N (s), z N s (s) − z N (s)z N y (s), z N s (s) .
No. 18, p. 17 where C depends on the constant of Estimate (3.7), that is, 1/4 √ 6C 1 and of the constants of immersions established in Proposition 2.2.

2 R 2 R Kdy ≤ 2 z N (s) 2 L 2 .
(z N ), such thatz N ⇀ z weak in L 2 0, S; H 2 (K) as N −→ ∞, z N → z strong in L 2 0, S; L 2 (K) as N −→ ∞, z N y → z y strong in L 2 0, S; L 2 (K) as N −→ ∞, z N s ⇀ z s weak in L 2 0, S; L 2 (K) as N −→ ∞.(3.9)From usual inequalities and Proposition 2.2 one hasR z N y (s)z N (s) − z y (s)z(s) Kdy = R z N y (s) − z y (s) z N (s) + z y (s) z N (s) − z(s) ∞ (R) z N y (s) − z y (s) 2 + 2 |z y (s)| 2 z N (s) − z(s) 2 ≤ C z N y (s) − z y (s) 2 + z N (s) − z(s)Taking (3.9) in this inequality, it yieldsS 0 z N y (s)z N (s) − z y (s)z(s) 2 ds ≤ C S 0 z N y (s) − z y (s) 2 + z N (s) − z(s) 2 ds −→ 0 as N −→ ∞.(3.10)Therefore, one hasz N y z N −→ z y z strong in L 2 0, S; L 2 (K) as N −→ ∞Thus, the proof of Theorem 3.2 is completed by using a similar argument as in Section 2.Finally, the uniqueness of solutions and the exponential decay rate of the energy are established in a similar way as in Section 2 EJQTDE, 2010 No. 18, p. 18 .