Abstract Generated abstract
The paper establishes existence and uniqueness results for the first boundary value problem for a one dimensional quasilinear parabolic equation with coefficients depending on time, space, the solution, and its spatial derivative. It weakens earlier growth assumptions on the ratio of the lower order term to the diffusion coefficient and removes sign restrictions on certain coefficient derivatives, using a comparison principle and a priori bounds for the solution and its derivatives. For a special class of equations, it also gives a necessary and sufficient integral condition for global solvability under arbitrary smooth initial and boundary data, and shows by example that a boundedness hypothesis in the comparison theorem cannot in general be omitted.
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MATHEMATICS
A. F. Filippov
ON CONDITIONS FOR THE EXISTENCE OF A SOLUTION OF A QUASILINEAR PARABOLIC EQUATION
(Presented by Academician I. G. Petrovskii on 3 VII 1961)
Existence theorems for the solution of the first boundary-value problem for the parabolic equation
\[ u_t=a(t,x,u,u_x)u_{xx}+b(t,x,u,u_x), \tag{1} \]
\[ u(0,x)=\varphi_0(x),\qquad u(t,X_1)=0,\qquad u(t,X_2)=0 \tag{2} \]
in the domain \(0\leq t\leq T,\ X_1\leq x\leq X_2\) (\(T\) arbitrary) were established in works \((^{1-4})\). In doing so, restrictions of three types were imposed on the functions \(a\) and \(b\): 1) restrictions on the growth rate of the ratio \(b/a\) as \(|u_x|\to\infty\); 2) smoothness of the functions \(a\) and \(b\); 3) the requirement that certain derivatives of the functions \(a\) and \(b\) (or certain combinations of them) have a definite sign. In the present note an existence theorem for a solution is established in which no restrictions of type 3) are imposed, and the restrictions of type 1) are weaker than in works \((^{1-4})\). In the case when the ratio \(b/a\) does not depend on \(t\) and \(x\), the imposed restriction on the growth rate admits no further weakening, since for equations of the form \(u_t=u_{xx}+u f(u_x)\) it is not only sufficient but also necessary for the existence of a solution under arbitrary boundary conditions.
Theorem 1. Let, in equation (1), the functions \(a\) and \(b\) be three times continuously differentiable, \(a(t,x,u,u_x)\geq a_0>0\),
\[ ub(t,x,u,0)\leq |u|\psi(|u|),\qquad \psi\geq 1,\qquad \int_0^\infty \frac{du}{\psi(u)}=\infty, \tag{3} \]
and for any \(U\), when \(|u|\leq U,\ |v|\geq V_0\), let
\[ |q(t,x,u,v)|\leq Cv^2,\qquad |q_v|\leq C_1|v|,\qquad |q_t|\leq C_2v^2,\qquad |q_x|\leq \alpha(v)|v|^3, \tag{4} \]
where \(q=b/a\); \(\alpha(v)\to 0\) as \(|v|\to\infty\); \(C,\ C_1,\ C_2,\ \alpha,\ V_0\) may depend on \(U\). Let the function \(\varphi_0\) in (2) satisfy the Hölder condition and the compatibility conditions
\[ \varphi_0(X_i)=0,\quad a(0,X_i,\varphi_0(X_i),\varphi_0'(X_i))\varphi_0''(X_i) +b(0,X_i,\varphi_0(X_i),\varphi_0'(X_i))=0, \]
\[ i=1,2. \]
Then the boundary-value problem (1), (2) in the domain \(D:\ (0\leq t\leq T,\ X_1\leq x\leq X_2)\), where \(T\) is arbitrary, has a unique solution in the class of functions continuous in \(D\) and possessing the derivatives \(u_t,\ u_x,\ u_{xx}\) inside \(D\). For this solution \(u_t,\ u_x,\ u_{xx}\) are continuous in \(D\) and satisfy the Hölder condition in \(t\) and \(x\); inside \(D\) there exist continuous \(u_{tt},\ u_{txx},\ u_{xxxx}\).
Remark. The theorem remains valid if condition (4) is replaced by any of the following two conditions:
\[ \text{A.}\qquad \left|\frac{\partial}{\partial v}\left(\frac{q}{v^2}\right)\right|\leq \frac{C_1}{|v|},\qquad |q_t|\leq C_2v^2\ln|v|,\qquad |q_x|\leq C_3v^2(\ln|v|)^{c_2}; \]
or
B. \(q=q(u,v)\) does not depend on \(t\) and \(x\), \(|q|\le |v|\psi(|v|)\), where the function \(\psi\) is the same as in (3).
Theorem 2. For the equation \(u_t=u_{xx}+uu_x\psi(u_x)\), where the function \(\psi\) has continuous \(\psi'''\) and \(|\psi(v)|\ge 1\), \(v\psi(v)>0\) for \(|v|\ge V_0\), the divergence of the integral
\[
\int_{\pm V_0}^{\pm\infty}\frac{dv}{\psi(v)}
\]
is a necessary and sufficient condition in order that, for arbitrary smooth initial and boundary conditions and any \(T\), the solution of the first boundary-value problem exist for \(0\le t\le T\).
Theorem 3. Let the functions \(u(t,x)\), \(v(t,x)\) be continuous in the bounded closed domain \(D_1\), lying in the strip \(0\le t\le T\), and let \(u_x, v_x\) and the left derivatives \(u_t^{-}, v_t^{-}, u_{xx}^{-}, v_{xx}^{-}\) exist everywhere inside \(D_1\). Let
\[
Lu\equiv u_t^{-}-f(t,x,u,u_x,u_{xx}^{-}),
\]
where the function \(f\) is nondecreasing in \(u_{xx}^{-}\); \(f\) and \(\partial f/\partial u\) are continuous. If: a) on the set \(\Gamma(\bar D_1)\) of those boundary points of the domain \(D_1\) at which \(t<T\), we have \(u\ge v\); b) inside \(D_1\), \(Lu\ge Lv\); c) \(u_x, u_{xx}^{-}\) (or \(v_x, v_{xx}^{-}\)) are bounded in \(D_1\), then \(u\ge v\) throughout the domain \(D_1\).
Remark. The following example shows that condition c) cannot be dropped. Let
\[
Lu\equiv u_t-9u_{xx}-2uu_x^3-bu_x^4(u^3-27x),
\]
\(D_1\) be the domain \(0\le t\le 1,\ 0\le x\le 1\); \(u=3x^{1/3}\), \(v=u+x^3(1-x)\xi e^{-\xi^2}\), \(\xi=xt^{-1/2}\). Then on \(\Gamma(\bar D_1)\), \(u=v\), while inside \(D_1\), \(Lu=0\), \(Lv<0\), if the number \(b\) is sufficiently large. However \(u<v\) inside \(D_1\).
Proof of Theorem 1. The uniqueness of the solution follows from Theorem 3. The existence of a solution on a sufficiently small time interval \(0\le t\le T_1\) follows from \((4\text{--}7)\). This solution, in the closed domain \(0\le t\le T_1,\ X_1\le x\le X_2\), has derivatives \(u_t, u_x, u_{xx}\) satisfying the Hölder condition in \(t\) and \(x\). Denote by \(T_2\) the upper bound of such values of \(T_1\). Then in the domain \(D_2\) \((0\le t<T_2,\ X_1\le x\le X_2)\) the solution exists and is continuous together with the derivatives \(u_t, u_x, u_{xx}\), while inside \(D_2\) there exist continuous \(u_{tt}, u_{txx}, u_{xxxx}\) \((^6)\).
We shall show that \(u, u_x, u_{xx}, u_t\) are bounded in \(D_2\). The boundedness of \(u\) is proved by comparison, by Theorem 3, of the solution \(u(t,x)\) of problem (1), (2) with the solution of the ordinary differential equation \(dz/dt=\psi(z)\). Thus, \(|u|\le M\) in the domain \(D_2\).
We shall prove the estimate of \(u_x\) on \(\Gamma(D_2)\), i.e., on that part of the boundary of the domain \(D_2\) where \(t<T_2\). Put
\[
U(t,x)=y(x-X_1-d)+M,
\]
where \(y(x)\) is the solution of the equation \(y''+y'\psi(y')=0\) with initial conditions \(y(0)=0,\ y'(0)=V_1\); the function \(\psi\) is such that \(|q|\le |v|\psi(|v|)\) for \(|v|\ge V_0\), \(\psi\ge 1\); the integral (3) diverges, \(V_1=\max\{V_0;\varphi_0'\}+1\), and the number \(d>0\) is such that \(y(-d)=-M\). Then for \(t=0\), for \(x=X_1\), and for \(x=X_3=\min\{X_1+d;X_2\}\) we have \(u(t,x)\le U(t,x)\), while for \(X_1<x<X_3\) we have \(0\le U\le M,\ V_1\le U_x\le y'(-d)\), \(LU>0,\ Lu=0\). By Theorem 3 we have \(u\le U\). Since \(u(t,X_1)=U(t,X_1)=0\), it follows that \(u_x(t,X_1)\le U_x(t,X_1)=y'(-d)\). Similarly, \(-u_x(t,X_1)\le y'(-d)\).
For estimating \(u_x\) inside the domain \(D_2\), note that if \(\varphi(t,x,u,v)=C\)—the first integral of the system
\[
\frac{dt}{0}=\frac{dx}{1}=\frac{du}{v}=-\frac{dv}{q(t,x,u,v)},
\tag{5}
\]
with \(\varphi_v>0\), then equation (1) is written in the form
\[
u_t=m(t,x,u,u_x)\frac{d}{dx}\varphi(t,x,u,u_x),
\]
where \(m=a/\varphi_v\).
Taking as the new unknown function \(p(t,x)=\varphi(t,x,\)
\(u(t,x), u_x(t,x))\), we obtain an equation of the form \(p_t=ap_{xx}+kp_x+\varphi_t\), where \(k\) is a continuous function. Now, in order to prove boundedness in \(D_2\) of the function \(p\), and then of the function \(v=u_x\), it is enough to require that \(v\) be bounded on \(\Gamma(D_2)\) (which has already been proved), and that the function \(\varphi\) in the region \(0\leq t<T_2,\ X_1\leq x\leq X_2,\ |u|\leq M,\ |v|\geq V^*\) (\(V^*\) is some number) have the following properties: a) \(\varphi\) is three times continuously differentiable; b) \(\varphi_v>0\); c) \(|\varphi_t|\leq \psi(|\varphi|)\) and the integral (3) diverges; d) as \(|v|\to\infty\) we have \(|\varphi|\to\infty\), and conversely.
We obtain such a function \(\varphi\) if, for every sufficiently large \(|C|\), we set \(\varphi(t,x,u,v)=C\) at the points of all integral curves of the system (5) satisfying the condition \(v=C\) when \(u=-M\) (in (5) we regard \(u\) as an independent variable, \(-M\leq u\leq M\)). From the inequality \(|q|\leq |v|\psi(|v|)\) it follows that \(\varphi\) has property d). To prove properties b) and c), we note that \(\varphi_v\) and \(\varphi_t\) may be regarded as derivatives of the solution of system (5) with respect to the initial conditions; therefore their estimate reduces to the estimate of the solution of some auxiliary system of linear differential equations. We obtain that, when condition (4) (or conditions A or B) is fulfilled, the function \(\varphi\) has properties b) and c).
After the boundedness of \(u\) and \(u_x\) has been proved, it is not difficult to prove the boundedness of \(u_{xx}\) and \(u_t\). From the boundedness of the latter it follows that \(u_x\) satisfies a Hölder condition in \(t\) and \(x\) throughout the whole region \(D_2\) (briefly: \(u_x\in H\)). Hence,
\(a(t,x,u(t,x),u_x(t,x))=A(t,x)\in H\),
\(b(t,x,u(t,x),u_x(t,x))=B(t,x)\in H\).
Since \(u_t=A(t,x)u_{xx}+B(t,x)\), by virtue of (5), \(u_t\in H,\ u_{xx}\in H\). Hence, if at \(t=T_2\) the function \(u\) is extended by continuity, then \(u(T_2,x)\) will satisfy a Hölder condition in \(x\); by virtue of the continuity of \(u_t,u_x,u_{xx}\), the compatibility conditions at the points \((T_2,X_1)\) and \((T_2,X_2)\) will be fulfilled. If \(T_2<T\), then with such initial conditions \(u=u(T_2,x)\), prescribed at \(t=T_2\), a solution \(u(t,x)\) for which \(u_t\in H,\ u_{xx}\in H\) exists also for \(T_2\leq t\leq T_2+\tau,\ \tau>0\). This contradicts the choice of the number \(T_2\). Hence \(T_2=T\). Theorem 1 is proved.
In Theorem 2, sufficiency follows from Theorem 1. To prove necessity, suppose that the integral referred to in the theorem converges. Denote by \(y(x)\) the solution of the equation \(y''+y'\psi(y')=0\) with initial conditions \(y(0)=0,\ y'(0)=+\infty\). Then, for sufficiently small \(h\), we have \(0<y<1,\ V_0<y'<\infty,\ y''<0\) for \(0<x\leq h;\ y'(h)>2\). Let \(m(t)\) be such a function that \(y'(m(t))=\dfrac{N}{1-t}\), \(N=y'(h)>2,\ 0<m(t)\leq h\) for \(0\leq t<1\). Put
\[ v(t,x)= \begin{cases} \dfrac{Nx}{1-t}, & \text{for } 0\leq x\leq c(t)=\dfrac{y(m(t))+N}{y'(m(t))},\\[6pt] y(x-c(t)+m(t))+N, & \text{for } c(t)\leq x\leq h. \end{cases} \]
Obviously, for \(0\leq x\leq h,\ 0\leq t<1\), the function \(v\) is continuously differentiable. It can be shown that \(v_t-v_{xx}-vv_x\psi(v_x)\leq 0\). Suppose that there exists a solution \(u(t,x)\), continuous for \(0\leq x\leq h,\ 0\leq t\leq 1\), of the equation of Theorem 2, for which \(u(t,0)=0\), and, at \(t=0\) and at \(x=h\), \(u(t,x)\geq v(t,x)\). Then, by Theorem 3, for arbitrarily small \(\delta>0\) and \(0\leq t\leq 1-\delta\), we must have \(u\geq v\). This contradicts the continuity of \(u(t,x)\), since \(u(1,0)=0\), whereas \(v(t,1-t)=N>2\).
Moscow State University
named after M. V. Lomonosov
Received
29 VI 1961
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