On viscous Burgers-like equations with linearly growing initial data

where ∂t = ∂/∂t. It is well-known that if u0 is bounded, (E) admits a unique global solution (cf. ). In this paper we consider the case that u0 is not bounded at the space infinity. This paper specifies the growth of nonlinear term as G(r) ∼ r for large r. A typical example is the viscous Burgers equation. Our goal is to solve the initial value problem when the initial data may grow linearly at the space infinity. We shall prove that the problem admits a unique local regular solution. The global existence is not expected in general even for n = 1 since u(x, t) = −x/(1 − t) is a solution of the viscous Burgers equation: ∂tu − ∆u + u∂xu = 0 with u0(x) = −x, where ∂x = ∂/∂x. We also obtain an optimal estimate of the existence time. In fact, the existence time interval (0, T ) is estimated from below by a constant multiple over a Lipschtz bound for initial data, T ≥ C2‖∇u0‖∞; here the constant C2 is estimated by the structure of G, and ‖∇u0‖∞ is defined by ‖∇u0‖∞ = (∑n i=1 ‖∂iu0‖∞ )1/2, where ∂iu0 = ∂u0/∂xi. To state our main result precisely we assume the following bounds for G = (G1, · · · , Gn) ∈ C(R;R) with some α ∈ (0, 1):


Introduction
We consider a viscous Burgers-like equation of the form where ∂ t = ∂/∂t.It is well-known that if u 0 is bounded, (E) admits a unique global solution (cf. [8]).In this paper we consider the case that u 0 is not bounded at the space infinity.This paper specifies the growth of nonlinear term as G(r) ∼ r 2 for large r.A typical example is the viscous Burgers equation.Our goal is to solve the initial value problem when the initial data may grow linearly at the space infinity.We shall prove that the problem admits a unique local regular solution.The global existence is not expected in general even for n = 1 since u(x, t) = −x/(1 − t) is a solution of the viscous Burgers equation: ∂ t u − ∆u + u∂ x u = 0 with u 0 (x) = −x, where ∂ x = ∂/∂x.We also obtain an optimal estimate of the existence time.In fact, the existence time interval (0, T ) is estimated from below by a constant multiple over a Lipschtz bound for initial data, T ≥ C 2 ∇u 0 ∞ ; here the constant C 2 is estimated by the structure of G, and ∇u 0 ∞ is defined by , where ∂ i u 0 = ∂u 0 /∂x i .
To state our main result precisely we assume the following bounds for G = (G 1 , • • • , G n ) ∈ C 2+α (R; R n ) with some α ∈ (0, 1): (C) < ∞, Here we set x = 1 + |x| 2 for x ∈ R n and G i is denotes the derivative of G i .
A typical example satisfying this assumption (C) is G i (r) = r 2 (1 ≤ i ≤ n).We prepare a few function spaces allowing growth at space infinity.Let L p m be of the form

Yoshikazu Giga And Kazuyuki Yamada
Of course, L p 0 = L p by definition so that • p,0 = • p .Let X B be of the form

Definition.
By a classical solution u of (E) we mean that u ∈ C(R n ×[0, T )) is C 2 in space and C 1 in time, and it solves (E).

Theorem (Existence and uniqueness of a solution of a viscous Burgers like equation).
Assume that The existence time estimate T ≥ T 0 is optimal in the sense that a classical solution may not exist in [0, T ) for T > T 0 .
Optimality is easily observed by the next example.

Example.
We set ξ, η ∈ R n and we take where "•" is the inner product.Then the function Remark.
(1) It is easy to see that this existence time estimate is invariant under a rotation of space variables x.If we do not care about rotation invariance of results, there is a sharper estimate for T by defining T 0 by where (2) If we consider ∂ t u − ε∆u + divG(u) = 0 for ε > 0 instead of the evolution equation (E), we still obtain the existence time estimate T ≥ T 0 independent of ε > 0. This is easily follows from our theorem by changing the variable t by s/ε or x by y/ √ ε.
For the viscous Burgers equation: the problem (E) is reduced to the initial value problem for the heat equation via the Hopf-Cole transformation.Indeed, we set We observe that v satisfies (B) We set w(x, t) = e 1 2 v(x,t) and observe that w satisfies the heat equation (The transformation form v to w is called the Hopf-Cole transformation.)Our problem is reduced to the unique solvability of the heat equation with initial data w ∼ e ax 2 for large x.The solvability and the existence time estimate is easily proved by the explicit solution formula.The uniqueness part is more subtle but it is widely studied for example in [10] .For the viscous Burgers equation our result easily follows from results for the heat equation [9] , [10] without a Lipschitz bound for u 0 .However, if n > 1 or G is general, this argument evidently fails to apply.
A classical result of Tychonov [9] states that the Cauchy problem for the heat equation has a unique classical solution in for a continuous initial data u 0 (x) satisfying growth condition for some positive constants C, a.Moreover, D. G. Aronson [1] generalized the result of Tychonov for a parabolic operator with variable coefficients with suitable conditions for A ij and A i for u 0 satisfying for some positive constant a.He proved that there is a unique solution in for Lu = 0 with u| t=0 = u 0 .K. Ishige [7] proved that solvability of Cauchy problem: for the initial data µ growing at space infinity.There are some more results for nonlinear equations (see e.g. [7], [3] ) but these results do not include (E).A recent paper [6] of A. Gladkov, M. Guedda and R. Kersner studied the unique solvability of with λ > 0, q > 1, when initial data v 0 is not necessary bounded.In fact, they proved that if u 0 (x) ≤ M 0 (α 0 + x 2 ) q/[2(q−1)]−γ with some positive constant M 0 , α, γ.Then there exists a unique local solution on R × (0, T ] provided that T satisfies If γ > 0, then the solution can be extended globally in time.When q = 2 the equation agrees with (B) .So their result qualitatively implies the local existence for the Burgers equation.However, in general their results do not overlap with ours.Like their result it is possible to prove the global existence when the growth order is less than linear.We shall discuss this topic in a forthcoming paper of the second author.Uniqueness of solutions without imposing growth conditions was recently studied by G. Barles, S. Biton and O. Ley [2] and K.-S Chou and Y.-C.Kwong [4] .
However, the class of quasilinear parabolic equations to which their theory applies excludes our equation (E).
Let us give the idea of the proof.If u 0 is bounded, (E) can be solved by the following iteration: But if u 0 is not bounded, it is difficult to solve (E) by the iteration (1).So we use another iteration: To use this iteration (2) it is necessary to study the solvability of the linear equation with growing coefficients in the transport term: Fortunately, it is not very difficult to solve the linear equation ( 3) for initial data v 0 ∈ BC, where BC is the set of all bounded continuous functions and BC m is defined by Estimating the heat kernel in (2), we get the estimate: Since u n+1 satisfies The maximum principle for (3) yields Applying the above maximum principle for v, we get

By the Gronwall inequality ∇u
for all k.By ( 4) and (5) we see that {u k } is a Cauchy sequence in L ∞ (0, T 0 − ε; L ∞ 1 ) for any ε ∈ (0, T ) so that u := lim k→∞ u k is solution of (E).It is easy to prove the uniqueness of solution of (E) by using the maximum principle for equation (3).
The key underlying estimate is an apriori estimate: for u of (E) which yields, by the Gronwall inequality (Lemma 3.1), a bound for ∇u(t) ∞ : It is natural to consider a linearly growing initial data for (E).We conclude this introduction by giving a formal argument to show that at most linearly growing initial data is allowed for existence of a solution.We postulate that u(x, t) = x α f (t) is a solution of (E).By (E) u must satisfy We observe that the growth of the left hand side is x α .By the assumption of G the growth of the right hand side is x 2α−1 .Hence α must satisfy α ≤ 2α − 1 so that α ≤ 1.

Estimates for the heat semigroup in weighted space
We recall several elementary properties of the heat kernel The next two lemmas are well-known but we give a proof for completeness.For a multi-index a = (a Lemma 2.1 (Derivatives of heat kernel).Derivatives of G t are of the form with some polynomial of x i and t −1 of the form Proof.
It is sufficient to prove in the case of a = (a 1 , 0, • • • , 0).
A standard induction argument yields Lemma 2.1 (In fact, p ai (x i , t) is a constant multiple of (4t) −ai (−1) n H ai ((4t) −1/2 x i ), where H ai is the Hermite polynomial defined by (e −s 2 ) (j) = (−1) j+1 H j (s)e −s 2 ) . 2 Lemma 2.2 (Polynomial multiplication).For a multi-index a the identity holds with some polynomial q j,i (t) of the form

Lemma 2.3 (Estimate of heat kernel in weighted space).
There is a constant Proof.An elementary calculation shows that with h i,z (y) defined by .
Clearly, we have sup We now calculate x −1 e t∆ f to get Estimating L ∞ -norm we obtain Here we have used A similar argument yields estimates of derivatives in weighted spaces.

Corollary 2.4.
There is a constant C = C(n, m, a) such that holds for all f ∈ L ∞ m and t > 0.
Lemma 2.5 ( Hölder continuity of the heat kernel in weighted space).There is a constant C = C(n, α) such that holds for all 0 < s ≤ t and 0 < α ≤ 1. Proof.
We set g m (x) = x −m .In a similar way of proving Lemma 2.3 we have, by In a similar way of proving Lemma 2.5, we obtain a more general version.
In this paper we use these estimates in finite time interval (0, T ) so we give the following version of the estimates in Corollary 2.4 and Corollary 2.6.
Here C T is a constant independent of f and t, s but may depend on T .

Gronwall type inequalities
In this section we recall several versions of Gronwall type inequalities.Lemma 3.1.
Assume that f ∈ L ∞ (0, T ), g ∈ L 1 (0, T ), satisfies f, g ≥ 0 a.e.t.Assume that h is a positive nondecreasing function on (0, ∞).Assume that c is a positive constant.Let H be a primitive function of 1/h.¿From now on we suppress the word "a.e.".Proof.We set Integrating this differential inequality, we get Since h is a positive, the function H is a monotone increasing function.Thus we conclude that (1) If H has the inverse, Lemma 3.1 implies In this paper we apply Lemma 3.1 when h(r) = r 2 and H(r) = r.If h(r) = r 2 , Lemma 3.1 implies that f satisfies (2) In Lemma 3.1 we assume f ≥ 0, h ≥ 0, c > 0. However, if h satisfies h(r) = r, it is not necessary to assume that f ≥ 0 and that c is a positive constant.Moreover we may take c as a function.
We shall state it for convenience.Assume that k ∈ L ∞ (0, T ), and g ∈ L 1 (0, T ) and g ≥ 0. Assume that f ∈ L ∞ (0, T ) satisfies This inequality is known as the famous Gronwall inequality and it is included in many standard text books.
We shall give an application of the Gronwall inequality.Lemma 3.3.
Assume that h ∈ L ∞ (0, T ) and that Assume that f, g ∈ L ∞ (0, T ) satisfy f, g ≥ 0 and that Proof.
By assumption Then for sufficiently small ε 0 > 0, w ε satisfies sup for all 0 < ε < ε 0 .Since w ε is negative at space infinity, w ε has a maximum point We are able to take ε small so that since the left hand side is finite by the assumption of p = (p 1 , . . ., p n ).Since (x ε , t ε ) is a maximum of w ε , we observe that for sufficiently small ρ > 0, where B ρ (x ε , t ε ) is a closed ball of radius ρ centered at (x ε , t ε ) ∈ R n × (0, T ).This contradicts the equation for w ε so we conclude that w ε ≤ 0. Sending ε to zero, we have v(x, t) ≤ 0, i.e.
A symmetric argument yields and the proof is now complete.2

Linear problem in a weighted space
We prove that solvability of a linear equation with growing coefficients at the space infinity.Definition.
By a mild solution of (L) we mean that u ∈ C(R n × [0, T )) satisfies Lemma 5.1 (Existence and uniqueness for bounded initial data).Assume that where BC is a set of bounded continuous functions.Let α ∈ (0, 1) and m ≥ 0 and assume that p i , ∂ i p i , q ∈ C α ((0, T ) : L ∞ m ).Then (L) has a unique classical solution u ∈ BC(R n × [0, T )). Proof.
Step. 1 (Construction of a mild solution).We construct a mild solution u ∈ BC(R n × [0, T )) for integral equation.
(a) Approximation.Let ψ ∈ C ∞ 0 (R n ) be a cut off function of the form satisfying and The unique existence of a classical solution is well-known [8] .By Remark 3.2.(2), and where g m is defined by the convergence is locally uniform.We shall prove t 0 e (t−s)∆ (divp l (s))u l (s)ds → t 0 e (t−s)∆ (divp(s))u(s)ds, t 0 e (t−s)∆ q(s)u l (s)ds → t 0 e (t−s)∆ q(s)u(s)ds, as l → ∞, * -weakly in L ∞ (R n × (0, T )).The first two convergences are easy to prove, so we only give a proof of the last convergence.We observe that •p l (y, s)u l (y, s)dyds)dxdt •p l (y, s)u l (y, s)dxdtdyds for all ϕ ∈ C ∞ 0 (R n × (0, T )).Since p l converge to p locally uniform, we see that ).We thus conclude that as l → ∞.This uniform bound for {u l } in (a) implies a bound for Since u l solves the approximate equation, by our convergence results we observe that the limit of u ∈ L ∞ (R n × (0, T )) satisfies Step. 2 (Regularity and continuity).
We shall prove the Hölder regularity and continuity for t > 0 and continuity at t = 0.By using Corollary 2.6 and the integral equation we see that u satisfies u ∈ C α ((0, T ); L ∞ 1 ) with α < 1/2.Since the initial data u 0 ∈ BC(R n ), it is easy to see that u 0 ∈ BU C 1 (R n ), here here BU C is the space of all bounded uniformly continuous functions.Since u solves the integral equation, and since e t∆ u 0 ∈ C([0, T ); BU C 1 ), we conclude that u ∈ C([0, T ); BU C 1 ).Thus u ∈ BC(R n × [0, T )).
Here we have invoked the Hölder regularity assumptions of p i , ∂ i p i , q.For further regularity see for instance [5] .If u ∈ BC(R n × [0, T )) is a classical solution of (L), then by the maximum principle (Lemma 4.1) we conclude the uniqueness of a solution.2 Remark 5.2.
It seems to be difficult to prove the uniqueness directly by estimating integral equation.
By the estimate for the heat semigroup in weighted space (Remark 2.7) we observe that u k+1 (t) ∞,1 Since {∂ j u k } satisfies the maximum principle (Lemma 4.1) for ∂ j u k implies that Multiplying both sides with ξ j ≥ 0 and taking the summation over j, we obtain We take the supremum of ξ = (ξ 1 , . . ., ξ n ), |ξ| = 1, where |ξ| = ( by the Schwarz inequality.We now apply Lemma 3.3 with Applying Lemma 3.3 with (7) inductivity, we conclude that ∇u k (t) ∞ ≤ k(t), t ∈ (0, T ) for all k ∈ N, provided that T < 1/C 2 ∇u 0 ∞ .By (6) we see that We again apply Lemma 3.3 with We shall estimate the deference: By definition w k satisfies We change the dependent variable by wk = w k x and observe that Therefore, We thus conclude that {u k } is Cauchy sequence in L ∞ ((0, T ); L ∞ 1 ).Let u be its limit.
Since u l converges to u locally uniformly and the ∇u l converges to ∇u in * -weak sense in L ∞ (R n × (0, T )).We see that  In other words u is a mild solution of (E).By Corollary 5.3 we observe that u is a classical solution and u ∈ C([0, T ); L ∞ 1 ).By the maximum principle (Lemma 4.1) it is easy to prove the uniqueness of a classical solution of (E).( By the way by construction we have ||∇u(t)|| ∞ ≤ k(t).However, this can be proved directly by estimating the integral equation and applying Remark 3.2(1).) 2 )h(f (s))ds for a.e.t ∈ (0, T ), then H(f (t)) − H(c) ≤ t 0 g(s)ds for a.e.t ∈ (0, T ).
ds , when c is positive.Of course, we may send c to zero in this case.If h(r) = r, Lemma 3.1 implies that f (t) ≤ ce t 0 g(s)ds .