UDC 517.947.5 : 517.947:45

ON THE PROPAGATION OF A PERTURBATION IN PROBLEMS CONNECTED WITH DEGENERATE QUASILINEAR EQUATIONS OF PARABOLIC TYPE

M. K. LIKHT

1. A number of applied problems, in particular problems of nonlinear heat conduction and filtration, are described by the equation

\[ \frac{\partial u}{\partial t}=\Delta f(u). \tag{1.1} \]

Ya. B. Zel’dovich and A. S. Kompaneets found [1] that if \(\dfrac{df}{du}=u^n\) \((n>0)\) and \(u(\bar{x},0)=0\), then the part of space to which the perturbation, introduced into the field at some point, has had time to propagate is finite at any moment of time.

An analogous fact was proved by G. I. Barenblatt and M. I. Vishik in [2] for the case where \(f(u)=\displaystyle\int_0^u u\Phi(u)\,du\), while \(\Phi(0)=0\) and \(\Phi'(u)>0\).

Degenerate quasilinear equations of parabolic type are considered in the most general form by O. A. Oleinik in [3], where a number of problems concerning the behavior of solutions of these equations are also formulated. In particular, the problem is posed there of studying the generalized solution in neighborhoods of the points at which degeneration occurs.

In the present note equation (1.1) is considered under certain general assumptions on \(f(u)\) and on the character of the perturbation at some point.

An estimate is refined for the distance traversed by the perturbation. In particular, for a bounded perturbation this distance, for large \(t\), grows no faster than \(\sqrt{t}\).

From the inequalities obtained in the work there also follow conclusions about the behavior of the solution near the points where it vanishes.

1. The equation describing the propagation of a plane wave has the form

\[ \frac{\partial u}{\partial t}=\frac{\partial^2 f(u)}{\partial x^2}. \tag{2.1} \]

In what follows it is always assumed: a) \(\dfrac{df}{du}\geqslant 0\) for all \(u\); b) in the domain \(D:\ x>0,\ t>0\) of the \(xot\)-plane, the functions \(u(x,t)\) and \(\dfrac{\partial f(u)}{\partial x}\) are continuous—

are; c) on the boundary of \(D\): \(u(x,0)=0\); \(u(0,t)=\psi(t)\), where \(\psi(t)\) is a nondecreasing function positive for \(t>0\); for every \(t>0\), \(\lim_{x\to\infty} u(x,t)=0\), and
\[ \lim_{x\to\infty}\frac{\partial f(u)}{\partial x}=0. \]

Without loss of generality, one may assume that \(f(0)=0\).

Lemma 1. A solution of equation (2.1) satisfying conditions a), b), and c) is nonnegative.

Proof. If on the line \(t=t_0\) the function \(u(x,t_0)\) has negative values, then it has a minimum at an interior point \(x_0\). At the same point \(f(u(x,t_0))\) has a minimum.

It follows from (2.1) that at this point
\[ \frac{\partial u}{\partial t}\geqslant 0. \]
Then below the plane \(P\):
\[ z=u(x_0,t_0)\frac{t+\varepsilon}{t_0+\varepsilon}\quad(\varepsilon>0), \]
for \(t<t_0\), there is a bounded part of the surface
\[ z=u(x,t). \]
This part of the surface has a supporting plane parallel to the plane \(P\).

At the point of support,
\[ \frac{\partial u}{\partial t}<0, \]
whereas
\[ \frac{\partial^2 f(u)}{\partial x^2}\geqslant 0, \]
which contradicts (2.1). Lemma 1 is proved.

1. Introduce the auxiliary function \(g(x,t)=f(u(x,t))\). Let \(f^{-1}(g)\) be the function inverse to \(f(u)\), and
   \[ \varphi(g)=\left.\frac{df}{du}\right|_{u=f^{-1}(g)}. \]
   The function \(g(x,t)\) satisfies the equation
   \[ \frac{\partial g}{dt}=\varphi(g)\frac{\partial^2 g}{\partial x^2} \tag{3.1} \]
   and the conditions
   \[ g(x,0)=0;\qquad g(0,t)=\psi_1(t), \]
   where \(\psi_1(t)=f(\psi(t))\) is a nonnegative nondecreasing function;
   \[ \lim_{x\to\infty} g(x,t)=0 \]
   for every \(t>0\).

Moreover, \(g(x,t)\geqslant 0\) in \(D\), is continuously differentiable with respect to \(x\) everywhere in \(D\);
\[ \frac{\partial g}{\partial x}\to 0\quad\text{as }x\to\infty \]
for every \(t>0\); \(\varphi(g)\) is a nonnegative function.

By \(G\) denote the surface over the plane \(xot\) given by the equation \(z=g(x,t)\).

Let \(P\) be a plane parallel to the axis \(ot\) and intersecting \(G\); \(G^+\) the closure of the part of \(G\) lying above \(P\), \(D^+\) its projection onto the plane \(xot\); \(G^-\) the part of \(G\) lying below \(P\), and \(D^-\) its projection onto the plane \(xot\).

If \(D''\) is part of some domain \(D'\) in the plane \(xot\) for which \(t<t_0\), then we shall say that \(D''\) is cut off from \(D'\) by the line \(t=t_0\).

Lemma 2. Every connected component of \(D^+\) is unbounded.

The proof of the lemma follows from the fact that, in the contrary case, by rotating a plane about the line of intersection of the plane \(P\) and the plane \(t=-\varepsilon\) \((\varepsilon>0)\), one can obtain a plane supporting the corresponding part of \(G\), and the point of support will be an interior point of \(D\). At this point
\[ \frac{\partial g}{\partial t}>0, \]
whereas
\[ \frac{\partial^2 g}{\partial x^2}<0, \]
which contradicts (3.1).

Lemma 3. If \(D'\) is cut off from \(D\) by the line \(t=t_0\), then every connected part of \(D'\) is either unbounded or has points located on the boundary of \(D\).

Proof. Suppose the contrary, and let \(D''\) be a connected bounded part of \(D'\) lying strictly inside \(D\).

Increasing \(t_1\) from zero, one can draw the line \(t=t_1\) in the \(xot\)-plane, supporting \(D''\). The point of support \(N\) will be an interior point of the outer boundary of \(D''\).

It is obvious that at the point \(N\)

\[ \frac{\partial g}{\partial t}\le 0,\qquad \frac{\partial^2 g}{\partial x^2}\ge 0, \]

but, in view of (3.1), this is possible only if

\[ \frac{\partial g}{\partial t}=0. \]

Consequently, the plane \(P\) at this point is tangent to the surface \(G\). Under a continuous displacement of the plane \(P\) downward, a curve \(\Gamma\) must be determined on the surface \(G\), at each point of which the plane tangent to the surface \(G\) is parallel to the plane \(P\), which is impossible.

Lemma 4. Everywhere in \(D\)

\[ \frac{\partial^2 g}{\partial x^2}\ge 0. \]

Proof. Suppose the contrary, and let \((x_0,t_0)\) be a point at which

\[ \frac{\partial^2 g}{\partial x^2}<0. \]

The plane \(t=t_0\) cuts the surface \(G\) along the curve \(\Gamma: z=g(x,t_0)\). Draw the tangent to \(\Gamma\) at the point \(x_0\), and through this tangent the plane \(P\), parallel to the axis \(ot\). On both sides of \(x_0\) there are points of \(\Gamma\) lying below the plane \(P\); however, by (3.1), for \(t<t_0\) there are points of the surface \(G\) situated above \(P\). Let \(\widetilde D^{+}\) be the connected component of \(D^{+}\) having points for \(t<t_0\) and the point \((x_0,t_0)\) on its boundary. By \(D_1^{-}\) denote the connected component of the part \(D^{-}\) cut off by the line \(t=t_0\), adjoining \(\Gamma\) to the left of \(x_0\), and by \(D_2^{-}\) the analogous part of \(D^{-}\), but adjoining \(\Gamma\) to the right of \(x_0\). By Lemma 2, the parts \(D_1^{-}\) and \(D_2^{-}\) cannot coincide.

The domain \(\widetilde D^{+}\) cannot intersect the line \(t=t_0\), for then one of the domains \(D_1^{-}\) or \(D_2^{-}\) would be bounded and separated from the boundary of \(D\) by the domain \(\widetilde D^{+}\). This can be reconciled with Lemma 2 only by assuming that \(P\) intersects the \(xot\)-plane to the right of \(x_0\). But, obviously, in this case \(\widetilde D^{+}\) merges into the part \(D^{+}\) intersecting the line \(t=t_0\).

A number of corollaries follow from Lemma 4.

Corollary 1. \(g(x,t)\) and \(u(x,t)\), for each fixed \(x\), are monotonically increasing functions of \(t\).

Corollary 2.

\[ \frac{\partial g}{\partial x}\le 0 \]

everywhere in \(D\) (similarly,

\[ \frac{\partial u}{\partial x} \]

).

Corollary 3. If \(x>x_0\) and \(t<t_0\), then

\[ g(x_0,t_0)\ge g(x,t) \tag{3.2} \]

(similarly for \(u(x,t)\)).

It is now not difficult to prove that the degeneracy of equation (2.1) is a necessary condition for a finite velocity of propagation of a perturbation.

Theorem 1. If, for \(u>0\),

\[ \frac{df}{du}\ge c>0, \]

then the solution of equation (2.1), satisfying conditions a), b), and c), is positive for all \(t>0\) and \(x>0\).

Proof. From Lemma 4 it follows that everywhere in \(D\)

\[ \frac{\partial g}{\partial t}\ge c\,\frac{\partial^2 g}{\partial x^2}. \tag{3.3} \]

Let \(v(x,t)\) be the solution of the equation

\[ \frac{\partial v}{\partial t}=c\,\frac{\partial^2 v}{\partial x^2}, \tag{3.4} \]

coinciding with \(g(x,t)\) for \(t=0\), \(x=0\), and \(x\to\infty\).

The function \(w=g-v\) vanishes on the boundary and satisfies inequality (3.3). Then, arguing in the same way as in the proof of Lemma 1, one can verify that \(w\ge 0\) everywhere in \(D\). But, as is known, \(v>0\) everywhere inside \(D\).

1. From the results proved in [3] it follows that the solution of equation (2.1), satisfying conditions a), b), and c), for each \(t>0\) becomes zero for sufficiently large \(x\).

By virtue of Corollary 2 one may assert that there exists a function \(x(t)\) separating the part of \(D\) in which \(u(x,t)\equiv 0\) from the part of \(D\) in which \(u(x,t)\ne 0\). Corollary 1 allows one to conclude that the function \(x(t)\) is nondecreasing.

The lemmas proved in the preceding paragraphs make it possible to obtain an estimate for the growth of \(x(t)\).

Theorem 2. If \(\dfrac{df}{du}\) is nondecreasing for \(u>0\) and, for every \(a>0\),

\[ \int_0^a \frac{[f'(\xi)]^{3/2}}{f(\xi)}\,d\xi < \infty, \tag{4.1} \]

then for the solution of equation (2.1), satisfying conditions a), b), and c), the inequality

\[ x(t)<2\sqrt{2t}\int_0^{\psi(t)} \frac{[f'(\xi)]^{3/2}}{f(\xi)}\,d\xi \tag{4.2} \]

holds.

Proof. We note that under the hypotheses of the theorem the function \(\varphi(g)\) is nondecreasing.

Let \(u(x_0,t_0)\ne 0\). Denote by \(D_0\) the domain in which \(t\le t_0\) and \(x>x_0\), and by \(\varphi_0\) the number \(\varphi(g(x_0,t_0))\). It is clear that in \(D_0\)

\[ \frac{\partial g}{\partial t}\le \varphi_0\,\frac{\partial^2 g}{\partial x^2}. \tag{4.3} \]

Integrating both sides of the inequality over the domain \(D_0\), we obtain

\[ \int_{x_0}^{x(t_0)} g(x,t_0)\,dx \le -\varphi_0\int_0^{t_0}\left.\frac{\partial g}{\partial x}\right|_{x=x_0}\,dt. \tag{4.4} \]

By virtue of the concavity of the function \(g(x,t)\), the integral on the left is not less than the area under the tangent to \(g(x,t_0)\) at \(x=x_0\), i.e.

\[ -\frac{1}{2}\,\frac{g_0^2}{\dfrac{\partial g_0}{\partial x}} \le -\varphi_0\int_0^{t_0}\left.\frac{\partial g}{\partial x}\right|_{x=x_0}\,dt, \tag{4.5} \]

where the zero subscript refers to the point \((x_0,t_0)\).

Dividing both sides of the inequality by \(\varphi_0\) and integrating it with respect to \(x_0\), we obtain

\[ -\frac{1}{2\varphi_0 \dfrac{\partial g_0}{\partial x}} \int_{x_0}^{x(t_0)} g^2(x,t_0)\,dx \leqslant g_0^2 t_0 . \tag{4.6} \]

And, finally, since \(g^2\) is also concave,

\[ g_0^2 \leqslant 2\varphi_0 \left(-\frac{\partial g_0}{\partial x}\right)^2 t_0 . \tag{4.7} \]

In what follows the zero subscript may be omitted, since the point \((x_0,t_0)\) was arbitrary, provided only that \(x_0 < x(t_0)\). We rewrite (4.7) in the form

\[ \frac{1}{\sqrt{t}} < -2\sqrt{2}\, \frac{\sqrt{\varphi(g)}}{g}\, \frac{\partial g}{\partial x}. \tag{4.8} \]

Integrating with respect to \(x\) from zero to \(x(t)\) and carrying out the corresponding changes of variables, we obtain what was required to be proved.

1. Let \(t(x)\) be the function inverse to \(x(t)\). It is clear that in the preceding arguments one could have integrated with respect to \(t\) not from zero, but from \(t=t(x_0)\). Repeating the transformations of item 4, in this case we obtain

\[ \frac{x(t)-x}{\sqrt{t-t(x)}} \leqslant 2\sqrt{2}\int_0^{u(x,t)} \frac{|f'(\xi)|^{3/2}}{f(\xi)}\,d\xi . \tag{5.1} \]

Inequality (5.1) makes it possible to draw certain conclusions about the nature of the function \(x(t)\). For example, the following assertion follows from it very simply.

Theorem 3. Under the assumptions of Theorem 2 the function \(x(t)\) is continuous, and for sufficiently small \(|t_1-t_2|\) the inequality

\[ |x(t_1)-x(t_2)| < |t_1-t_2|^{1/2} \tag{5.2} \]

holds.

Estimates of the function \(u(x,t)\) in a neighborhood of the points of degeneration are obtained no less simply from (5.1). For example, the following assertion is valid.

Theorem 4. Suppose that, under the assumptions of Theorem 2,

\[ \frac{df}{du}=cu^n+o(u^n)\quad (n>0). \]

If, for a given \(t\), \(x'(t)\) exists, then, up to small quantities of higher order,

\[ u(x,t) > \left[ \frac{x'^2 n^2}{32c(n+1)^2} \right]^{1/n} [t-t(x)]^{1/n}. \tag{5.3} \]

From the examples in [1, 2] one can see that inequality (5.3) is sharp with respect to the orders of smallness of the quantities.

References

1. Zel’dovich Ya. B., Kompaneets A. S. On the theory of heat propagation with thermal conductivity depending on temperature. Collection dedicated to the 70th anniversary of A. F. Ioffe. Moscow, 1950.

2. Barenblatt G. I., Vishik M. I. PMM, 20, issue 3, 411–417, 1956.

3. Oleinik O. A. On certain nonlinear problems in the theory of partial differential equations. First Summer Mathematical School, part II. Kiev, “Naukova dumka,” 1964.

Received by the editors
July 6, 1965

Giprostal’, Kharkov