MATHEMATICS

A. M. Il’in and O. A. Oleinik

ON THE BEHAVIOR OF SOLUTIONS OF THE CAUCHY PROBLEM FOR SOME QUASILINEAR EQUATIONS AS TIME INCREASES WITHOUT BOUND

(Presented by Academician I. G. Petrovskii, 10 I 1958)

The paper investigates the behavior of solutions of the Cauchy problem for a quasilinear parabolic equation with two independent variables, and also the behavior of generalized solutions of the Cauchy problem for a quasilinear first-order equation as (t \to \infty).* We shall restrict ourselves to considering equations of the form

[
\frac{\partial u}{\partial t}+\frac{\partial \varphi(u)}{\partial x}
=\varepsilon \frac{\partial^2 u}{\partial x^2},\qquad \varepsilon>0
\tag{1}
]

[
\frac{\partial u}{\partial t}+\frac{\partial \varphi(u)}{\partial x}=0,
\tag{2}
]

which are model equations for gas dynamics equations. The question of the behavior of the solution of the Cauchy problem as (t \to \infty) for an equation of the form (\partial u/\partial t=\varepsilon\,\partial^2 u/\partial x^2+F(u)) was considered in ((^1)). For equations (1) and (2), with (\varphi(u)=u^2/2) and with an initial function summable on the whole axis, this question was studied in ((^2)).

We shall consider solutions of the Cauchy problem for equations (1) and (2) in the half-plane (t \ge 0) with initial condition

[
u\big|_{t=0}=u_0(x),\qquad -\infty<x<+\infty,
\tag{3}
]

where (u_0(x)) is a bounded measurable function. The proof of existence, uniqueness, and properties of these solutions is given in ((^3)).

1. We shall assume that (\varphi(u)) has continuous derivatives up to the fourth order, (\varphi'' \ge \mu>0), and that (u_0(x)\to u_-) as (x\to-\infty) and (u_0(x)\to u_+) as (x\to+\infty). It has been proved that there exists in (R{t\ge0,\,-\infty<x<+\infty}) a bounded function (u_\varepsilon(t,x)), satisfying equation (1) for (t>0) and assuming the initial values (3) in the weak sense (see ((^3)), p. 35).

Theorem 1. Let (u_->u_+), and suppose that for (u_0(x)) the integrals

[
\int_{-\infty}^{0}\bigl(u_0(x)-u_-\bigr)\,dx,\qquad
\int_{0}^{+\infty}\bigl(u_0(x)-u_+\bigr)\,dx
\tag{4'}
]

exist and their sum is equal to (A). Equation (1) has a unique solution (\widetilde u_\varepsilon(x-Kt)), depending only on (x-Kt), where (K=[\varphi(u_+)-\varphi(u_-)]/[u_+-u_-]), and such that

[
\int_{-\infty}^{0}\bigl(\widetilde u_\varepsilon(x)-u_-\bigr)\,dx
+
\int_{0}^{+\infty}\bigl(\widetilde u_\varepsilon(x)-u_+\bigr)\,dx
=A.
]

* This problem was brought to our attention by I. M. Gel'fand.

As (t\to\infty)
[
\left|\widetilde u_\varepsilon(x-Kt)-u_\varepsilon(t,x)\right|\to 0
]
uniformly with respect to (x). If, for (u_0(x)), the additional conditions
[
\left|\int_{-\infty}^{x}(u_0(x)-u_-)\,dx\right|\leq M_1e^{\alpha_1x},\qquad
\left|\int_{x}^{+\infty}(u_0(x)-u_+)\,dx\right|\leq M_1e^{-\alpha_1x}
\tag{4}
]
are satisfied for some constants (\alpha_1>0) and (M_1>0), then
[
\left|\widetilde u_\varepsilon(x-Kt)-u_\varepsilon(t,x)\right|\leq M_2e^{-\beta t}
]
for all (x) and (t), where (\beta>0), (M_2>0) are certain constants.

First we shall prove Theorem 1 for a twice continuously differentiable function (u_0(x)), under the condition that (u'0(x)\to0) as (x\to\pm\infty) and (u''_0(x)) is bounded. In this case, as shown in (4), (u\varepsilon(t,x)) is continuous and bounded in (R), together with all derivatives entering equation (1), and for (t>0) this equation may be differentiated with respect to (x). It suffices to prove Theorem 1 for (K=0). Indeed, make the change of variables (x-Kt=x_1,\ t=t_1). In the new variables equation (1) is written in the form
[
\partial u/\partial t_1+\partial\varphi_1(u)/\partial x_1
=\varepsilon\,\partial^2u/\partial x_1^2,
]
where (\varphi_1(u)=\varphi(u)-Ku), and consequently (\varphi_1(u_+)=\varphi_1(u_-)) and (K_1=0). The proof of Theorem 1 under these assumptions is based on the following lemmas.

Lemma 1. Let (u(t,x)) be a continuous function in (R), having for (t>0) the derivatives (u_x, u_t, u_{xx}), and suppose that when (u(t,x)\leq0) the condition
[
\varepsilon\,\partial^2u/\partial x^2-\partial u/\partial t
+a(t,x,u)\,\partial u/\partial x+c(t,x,u)u\leq0
]
is satisfied, where (a) and (c) are bounded functions and (c\leq0) for (u\leq0). If (u(0,x)\geq0) and (u(t,x)\geq M(t)\sqrt{x^2+1}), where (M(t)) is a continuous function, then (u(t,x)\geq0) in (R).

With the aid of this lemma, Lemmas 2 and 3 are established.

Lemma 2. As (x\to\pm\infty), the solution (u_\varepsilon(t,x)) of problem (1), (3) tends respectively to (u_+) and (u_-), uniformly on each finite interval of variation of (t).

Lemma 3. As (x\to\pm\infty), (\partial u_\varepsilon/\partial x\to0) uniformly on each finite interval of variation of (t).

Lemma 4. For any (t\geq0) and for any (x), the integrals
[
\int_{-\infty}^{x}(u_\varepsilon(t,x)-u_-)\,dx,\qquad
\int_{x}^{+\infty}(u_\varepsilon(t,x)-u_+)\,dx
]
exist.

The proof of this assertion, for example for the first integral, is based on the fact that the function (v_\varepsilon(t,x)), which satisfies in (R) the equation
[
\partial v/\partial t+\varphi(u)-\varphi(u_+)
=\varepsilon\,\partial^2v/\partial x^2
]
and the initial condition
[
v_\varepsilon(0,x)=\int_{-\infty}^{x}(u_0(x)-u_-)\,dx,
]
for any (t\geq0) tends to zero as (x\to-\infty), and that
[
\partial v_\varepsilon/\partial x=u_\varepsilon(t,x)-u_-.
]

Lemma 5. For any (t\geq0),
[
\int_{-\infty}^{0}(u_\varepsilon(t,x)-u_-)\,dx
+\int_{0}^{+\infty}(u_\varepsilon(t,x)-u_+)\,dx=A.
]

This equality is established by differentiating its left-hand side with respect to (t), taking into account equation (1) and Lemmas 2 and 3.

Lemma 6. As (x\to-\infty), (u_\varepsilon(t,x)\to u_-), and as (x\to+\infty), (u_\varepsilon(t,x)\to u_+), uniformly in (t) and (\varepsilon) ((t\geq0,\ 0<\varepsilon\leq1)).

To prove this lemma, it is first established that the functions
[
v_-(t,x)=\int_{-\infty}^{x}(u_\varepsilon(t,x)-u_-)\,dx
\quad\text{and}\quad
v_+(t,x)=\int_{x}^{+\infty}(u_\varepsilon(t,x)-u_+)\,dx
]
uniform-

but as (t) and (\varepsilon) tend to zero as (x \to \pm \infty). In this case (v_{-}(t,x)) is estimated from below by the function
[
w_1=\int_{-\infty}^{x}\bigl(\tilde u_{\varepsilon}(x+c)-u_{-}\bigr)\,dx-\delta,
]
where (c) is a sufficiently large constant, and from above by the function
[
w_2=M_3 e^{\alpha_3 x}+\delta,
]
where (M_3>0) and (\alpha_3>0) are certain constants, (\delta>0) is an arbitrarily small number, since (v_{-}-w_1) and (w_2-v_{-}) satisfy equations to which the maximum principle is applicable. For any (\varepsilon>0), (\partial u_{\varepsilon}/\partial x<c_1), and therefore Lemma 6 follows from the assertion just stated for (v_{+}) and (v_{-}).

To prove Theorem 1 for (K=0), using the preceding lemmas, it is established that
[
z_{\varepsilon}(t,x)=\int_{-\infty}^{x}\bigl(u_{\varepsilon}(t,x)-\tilde u_{\varepsilon}(x)\bigr)\,dx
]
satisfies an equation for which the maximum principle is valid, and for all (x) and (t \ge 0) the inequality
[
|z_{\varepsilon}(t,x)|\leq \delta+M e^{-\beta t-\alpha\theta(x)},
\tag{5}
]
holds, where (M,\beta,\alpha) are certain positive numbers depending on (\delta); (\delta>0) is an arbitrarily small number; and (\theta(x)>0) is a suitably chosen function. If conditions (4) are satisfied, then (5) is valid with (\delta=0) and for certain (M,\alpha,\beta). Since (\partial(u_{\varepsilon}-\tilde u_{\varepsilon})/\partial x<c_2) for all (t), the assertion of Theorem 1 follows from (5) for a smooth function (u_0(x)) satisfying the conditions stated above. To prove Theorem 1 for an arbitrary bounded measurable function (u_0(x)), we construct a sequence of smooth functions (u_0^n(x)), satisfying the conditions indicated earlier, and such that
[
\int_{-\infty}^{+\infty}|u_0(x)-u_0^n(x)|\,dx\to 0
]
as (n\to\infty). Taking into account (3), that for the corresponding solutions of the Cauchy problem
[
\int_{-\infty}^{+\infty}|u_{\varepsilon}(t,x)-u_{\varepsilon}^n(t,x)|\,dx\to 0
]
as (n\to\infty) and for any (t), and using the already proved Theorem 1 for the functions (u_0^n(x)), we obtain the complete proof of Theorem 1.

As examples show, the existence of the integrals ((4')) is an essential condition for the validity of Theorem 1.

Theorem 2. Suppose (u_{+}=u_{-}=a). As (t\to\infty), (|u_{\varepsilon}(t,x)-a|\to 0) uniformly for all (x). If
[
|u_0(x)-a|\leq M_1 e^{-\alpha_1|x|}
]
for some (\alpha_1>0) and (M_1>0), then for all (x) and (t\ge 0)
[
|u_{\varepsilon}(t,x)-a|\leq M t^{1/2}|\ln t|^{\beta},
\tag{6}
]
where (M>0) and (\beta>0) are constants.

To prove Theorem 2 we use the following proposition: if (u_0(x)\geq u_0^1(x)), then for the corresponding solutions of the Cauchy problem (1), (3) the inequality (u_{\varepsilon}(t,x)\geq u_{\varepsilon}^1(t,x)) holds. Let (u_0^1(x)=a+2\delta) for (x\leq -N), (u_0^1(x)=a+\delta) for (x\geq N), and (u_0^1(x)\geq u_0(x)) for all (x). Then (u_{\varepsilon}(t,x)\leq u_{\varepsilon}^1(t,x)). But, by Theorem 1, (|u_{\varepsilon}^1(t,x)-a|\leq 3\delta), if (t\geq T) and (T) is sufficiently large. Consequently, for (t\geq T), (u_{\varepsilon}(t,x)\leq a+3\delta). In the same way we obtain an estimate from below. Estimate (6) is established on the basis of studying the dependence of (T) on (\delta).

Theorem 3. Suppose (u_{+}>u_{-}), and (H(s)) is the function defined by the equality (s=\varphi'(H(s))) for (\varphi'(u_{-})\leq s\leq \varphi'(u_{+})), (H(s)=u_{-}) for (s\leq \varphi'(u_{-})), and (H(s)=u_{+}) for (s\geq \varphi'(u_{+})). Then the solution (u_{\varepsilon}(t,x)) of problem (1), (3) tends uniformly in (x) and (\varepsilon) to (H(x/t)) as (t\to\infty).

To prove Theorem 3 the following lemma is established:

Lemma 7. For any (\delta>0) there exists (N(\delta)>0) such that
[
|u_{\varepsilon}(t,x)-u_{-}|\leq \delta
]
for
[
x-\varphi'(u_{-})t\leq -N,
]
and
[
|u_{\varepsilon}(t,x)-u_{+}|\leq \delta
]
for
[
x-\varphi'(u_{+})t\geq N.
]

Using this lemma, the assertion of Theorem 3 for
(\varphi'(u_-)\ll x/t \ll \varphi'(u_+)) is easily obtained by considering the equation satisfied by (u_\varepsilon(t,x)-H(x/t)).

1. Consider the generalized solution (u(t,x)) of the Cauchy problem for (t\geqslant0) for equation (2) with initial condition (3). It has been proved (5) that (u_\varepsilon(t,x)\to u(t,x)) as (\varepsilon\to0) at every point of continuity of (u(t,x)), and that the set of discontinuity points of (u(t,x)) on each straight line (t=t_0>0) is at most countable (see also (3)).

Theorem 4. Let (u_->u_+),

[
\int_{-\infty}^{0} (u_0(x)-u_-)\,dx+
\int_{0}^{+\infty} (u_0(x)-u_+)\,dx=A,
]

(u(t,x)) be the solution of problem (2), (3). Let (\sigma(x)=u_-) for (x\leqslant x_0) and (\sigma(x)=u_+) for (x\geqslant x_0), where (x_0) is determined so that

[
\int_{-\infty}^{0} (\sigma(x)-u_-)\,dx+
\int_{0}^{+\infty} (\sigma(x)-u_+)\,dx=A.
]

Then, as (t\to\infty), (|\sigma(x-Kt)-u(t,x)|\to0) uniformly in (x) outside the region
(x_0-\delta\ll x-Kt\ll x_0+\delta), where (\delta>0) is an arbitrary number,
(K=[\varphi(u_+)-\varphi(u_-)]/[u_+-u_-]).

This theorem is established with the aid of Lemmas 5, 6 and certain properties of (u(t,x)) given in § 6 of paper (3).

Theorem 5. If (u_->u_+) and (u_0(x)=u_-) for (x<-N) and (u_0(x)=u_+) for (x>N), where (N>0) is some number, then (u(t,x)) coincides with the function (\sigma(x-Kt)), defined in Theorem 4, for (t\geqslant T), where (T) is some number.

Theorem 6. Let (u_+=u_-=a). Then (|u(t,x)-a|\to0) as (t\to\infty) uniformly in (x).

This theorem is proved with the aid of Theorem 5 analogously to the way Theorem 2 was proved with the aid of Theorem 1.

Theorem 7. Let (u_-