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MATHEMATICS
T. D. VENTTSEL’
ON SOME QUASILINEAR PARABOLIC SYSTEMS WITH GROWING COEFFICIENTS
(Presented by Academician I. G. Petrovskii, 4 V 1961)
In paper (¹) the existence of a solution of the first boundary-value problem and of the Cauchy problem in the large was proved for quasilinear parabolic systems of the form
\[ \begin{gathered} \frac{\partial^2 u_i}{\partial x^2} = \frac{\partial u_i}{\partial t} + \sum b_{ij}(x,t,u)\frac{\partial u_j}{\partial x} + f_i(x,t,u), \\ u=(u_1,\ldots,u_N),\qquad i=1,\ldots,N, \end{gathered} \tag{1} \]
with bounded coefficients \(b_{ij}\) (and under certain assumptions concerning the functions \(f_i\)).
Set
\[ B(M)=\max_{|u_i|\le M}|b_{ij}(x,t,u)|. \]
A small modification of the proof of the existence theorem in (¹) makes it possible to prove the existence of a solution of the first boundary-value problem and of the Cauchy problem in the large when
\[ \int^\infty \frac{dM}{M B^2(M)}=\infty . \]
Analogous theorems for the Cauchy problem were proved in the paper of S. D. Eidel’man (²).
In the present paper systems of the form
\[ \begin{aligned} \varepsilon \frac{\partial^2 u}{\partial x^2} &= \frac{\partial u}{\partial t} + \frac{\partial \varphi(u,v)}{\partial x}, \\ \varepsilon \frac{\partial^2 v}{\partial x^2} &= \frac{\partial v}{\partial t} + \frac{\partial \psi(u,v)}{\partial x} \end{aligned} \tag{2} \]
are considered.
For such systems, in some cases it is possible to prove existence theorems in the large even when the coefficients of \(\partial u/\partial x\) and \(\partial v/\partial x\) have more rapid growth in \(u\) and \(v\).
In paper (¹) (Theorem 4) it was proved that the solution of the first boundary-value problem for any system of the form (2) with sufficiently smooth \(\varphi,\psi\) and boundary functions exists for all values of \(t\), if there is an a priori estimate for \(u\) and \(v\) in \(C\). In the present paper, for solutions of certain systems of the form (2), an a priori estimate of a different character is established, and it is then indicated for which systems boundedness of \(u\) and \(v\) in \(C\) follows from this estimate.
For system (2) the first boundary-value problem is considered with the conditions
\[ u\big|_{t=0}=u_0(x),\qquad v\big|_{t=0}=v_0(x), \tag{3} \]
\[ u\big|_{x=x_1}=u\big|_{x=x_2}=0,\qquad v\big|_{x=x_1}=v\big|_{x=x_2}=0. \]
Everywhere in what follows it is assumed that the first-order system corresponding to (2) as \(\varepsilon\to 0\) is of hyperbolic type.
Consider the equation
\[ \varphi_v F_{uu}-(\varphi_u-\psi_v)F_{uv}-\psi_u F_{vv}=0. \tag{4} \]
This equation has the same type as system (2) for \(\varepsilon=0\). Indeed, the type of equation (4) is determined by the sign of the expression
\[ (\varphi_u-\psi_v)^2+4\varphi_v\psi_u. \tag{5} \]
and the type of the system by the roots of the equation
\[ \lambda^2-\lambda(\varphi_u+\psi_v)+(\varphi_u\psi_v-\psi_u\varphi_v)=0, \]
whose discriminant coincides with (5).
Theorem 1. If equation (4) has a solution \(F(u,v)\) such that, for all \(u,v\),
\[ F_{uu}\xi^2+2F_{uv}\xi\eta+F_{vv}\eta^2 \geq \mu(u,v)(\xi^2+\eta^2),\qquad \mu>0, \tag{6} \]
then
\[ \int_{x_1}^{x_2} F(u(x,T),v(x,T))\,dx \leq \int_{x_1}^{x_2} F(u_0(x),v_0(x))\,dx. \tag{7} \]
Proof. Multiply the equations of system (2) by \(F_u, F_v\), respectively, add, and integrate over the rectangle
\(R\{x_1\leq x\leq x_2,\;0\leq t\leq T\}\); the integral over \(R\) will be denoted by \([\cdot]\). We have
\[ \varepsilon\left[ F_u\frac{\partial^2 u}{\partial x^2} + F_v\frac{\partial^2 v}{\partial x^2} \right] = \left[\frac{\partial F}{\partial t}\right] + \left[ F_u\left(\varphi_u\frac{\partial u}{\partial x} +\varphi_v\frac{\partial v}{\partial x}\right) + F_v\left(\psi_u\frac{\partial u}{\partial x} +\psi_v\frac{\partial v}{\partial x}\right) \right]. \tag{8} \]
Since the function \(F(u,v)\) is a solution of equation (4), the last term in (8) has the form \(\partial G(u,v)/\partial x\).
By adding to \(F(u,v)\) a suitably chosen linear function, one can ensure that the conditions
\[ F(0,0)=F_u(0,0)=F_v(0,0)=0 \tag{9} \]
are satisfied.
Integration by parts gives
\[ -\varepsilon\left[ F_{uu}\left(\frac{\partial u}{\partial x}\right)^2 + 2F_{uv}\frac{\partial u}{\partial x}\frac{\partial v}{\partial x} + F_{vv}\left(\frac{\partial v}{\partial x}\right)^2 \right] = \int_{x_1}^{x_2} F\big|_{t=T}\,dx - \int_{x_1}^{x_2} F\big|_{t=0}\,dx. \tag{10} \]
(The integral of \(\partial G/\partial x\) vanishes by virtue of the boundary conditions.) Since the function \(F\) is convex, (7) follows from (10).
Introduce the notation:
\[ B(M)= \max_{\substack{x,t\in R\\ |u|,\,|v|\leq M}} \left(|\varphi_u|,\;|\varphi_v|,\;|\psi_u|,\;|\psi_v|\right), \qquad B_1(M)= \max_{\substack{x,t\in R\\ |u|,\,|v|\leq M}} \left(|D^2\varphi|,\;|D^2\psi|\right) \]
(\(D^2\) is any second derivative),
\[ f(|u|)=\min\left(\min_v F(u,v),\;\min_v F(-u,v)\right), \]
\[ g(|v|)=\min\left(\min_u F(u,v),\;\min_u F(u,-v)\right). \]
Theorem 2. Let the coefficients of system (2) and the boundary functions satisfy the smoothness conditions formulated in Theorem 1 of paper (1), and let equation (4) have a solution \(F(u,v)\) satisfying conditions (6), (9) and such that
\[ B(M)+MB_1(M)=o\left(f\left(\frac{M}{2}\right)\right), \]
\[ B(M)+MB_1(M)=o\left(g\left(\frac{M}{2}\right)\right). \tag{11} \]
Then the solution of the first boundary-value problem (2), (3) exists for all \(t\).
Proof. As was already stated, to prove the existence theorem it is enough to obtain an a priori estimate in \(C\) for \(u\) and \(v\). Let
\[ M=\max_{x,t\in R}(|u|,|v|),\qquad M_1=\max_{x,t\in R}\left(\left|\frac{\partial u}{\partial x}\right|,\left|\frac{\partial v}{\partial x}\right|\right). \]
From the estimates ob-
of [1] (they are set out in detail in [3]), it follows that
\[ M_1 \leq K\bigl(MB(M)+M^2B_1(M)\bigr). \tag{12} \]
Let now, for definiteness, \(M=\max |u|\). From estimate (7) it follows that
\[ \int_{x_1}^{x_2} f(|u|)\big|_{t=T}\,dx \leq \int_{x_1}^{x_2} F(u_0(x),v_0(x))\,dx=A. \tag{13} \]
Let \(E\) be the set on the interval \([x_1,x_2]\) where \(M/2\leq |u(x,T)|\leq M\). By virtue of (13), \(\operatorname{mes} E\leq A/f(M/2)\), whence it follows that
\[ M_1 \geq \max\left|\frac{\partial u}{\partial x}\right| \geq \frac{Mf(M/2)}{A}. \tag{14} \]
The estimates (12), (14) give
\[ \frac{Mf(M/2)}{A} \leq M_1 \leq K\bigl(MB(M)+M^2B_1(M)\bigr). \tag{15} \]
If \(M=\max |v|\), we similarly obtain
\[ \frac{Mg(M/2)}{A} \leq M_1 \leq K\bigl(MB(M)+M^2B_1(M)\bigr). \tag{16} \]
From condition (11) and either of these estimates, the boundedness of \(M\) follows.
We shall say that a function \(\varphi(s)\) grows like \(|s|^p\) if
\[ k_1|s|^p \leq |\varphi(s)| \leq k_2(1+|s|^p). \]
Theorem 3. Let \(\varphi=\varphi(v)\), \(\psi=\psi(u)\), and suppose that \(\varphi(s)\) and \(\psi(s)\) grow like \(|s|^p\), and that the order of growth is lowered by 1 under differentiation and raised by 1 under integration of these functions. Then, if the coefficients of system (2) and the boundary functions (3) satisfy the smoothness conditions of Theorem 1 of [1], the solution of problem (2), (3) exists for all \(t\).
Proof. Equation (4) in this case has the form
\[ \varphi'(v)F_{uu}=\psi'(u)F_{vv}, \]
and a convex solution of it is the function
\[ F(u,v)=\int_0^u \psi(u)\,du+\int_0^v \varphi(v)\,dv. \]
Indeed, by virtue of the hyperbolicity of (2) for \(\varepsilon=0\), \(\varphi'(v)\) and \(\psi'(u)\) have the same sign, and without loss of generality they may be regarded as positive. Conditions (9) are satisfied, since one can always assume that \(\varphi'(0)=\psi'(0)=0\).
We have
\[ f(|u|)=\min\left(\int_0^u \psi(u)\,du,\ \int_{-u}^0 \psi(u)\,du\right), \]
\[ g(|v|)=\min\left(\int_0^v \varphi(v)\,dv,\ \int_{-v}^0 \varphi(v)\,dv\right), \]
i.e. \(f(M)\) and \(g(M)\) grow like \(M^{p+1}\). The function \(F(M)+MB_1(M)\) grows like \(M^{p-1}\), whence estimate (11) follows. Consequently, problem (2), (3) satisfies the conditions of Theorem 2 and has a solution for all \(t\).
We note that estimate (7) does not depend on \(\varepsilon\). The estimate for \(M=\max(|u|,|v|)\), which is obtained from (11), (15), (16), depends on \(\varepsilon\), since the constant \(K\) in (12) depends on \(\varepsilon\).
Moscow State University
named after M. V. Lomonosov
Received
25 IV 1961
REFERENCES
- T. D. Venttsel, DAN, 117, No. (1957).
- S. D. Eidelman, Matem. sborn., 53, No. 1 (1961).
- T. D. Venttsel, Dissertation, Moscow, 1958.