Full Text
A. V. Grekov
THE DIRICHLET PROBLEM FOR CERTAIN QUASILINEAR PARABOLIC EQUATIONS
(Presented by Academician V. I. Smirnov on 18 V 1960)
The subject of the present note is the proof of existence and uniqueness theorems for the solution of the Dirichlet problem for certain equations of parabolic type of the second order containing functions of two independent variables.
In the plane \((x,t)\) consider a curvilinear trapezoid \(D\), bounded by the segments of the straight lines \(t=0\), \(t=T\) \((T>0)\) and by the curves \(x_1=\varphi_1(t)\), \(x_2=\varphi_2(t)\), where \(\varphi_1,\varphi_2\) are continuous functions and \(\varphi_1(t)<\varphi_2(t)\) for \(t\in[0,T]\). The part of the boundary \(D\) consisting of the segment of the straight line \(t=0\) and the curves \(\varphi_1(t)\), \(\varphi_2(t)\), will be denoted by \(\Gamma\). Let \(\overline D\) be the closure of \(D\). Suppose, further, that the function \(u(x,t)\) is defined in \(\overline D\) and solves the following problem
\[ L_{xt}u\equiv u_t-a(x,t)u_{xx}+b(x,t)u_x+c(x,t)u+f(x,t)=0 \quad \text{for } x,t\in \overline D\setminus\Gamma; \tag{1} \]
\[ u|_{t=0}=\varphi(x) \quad \text{for } x\in[\varphi_1(0),\varphi_2(0)]; \]
\[ u|_{x=\varphi_i(t)}=\psi_i(t) \quad \text{for } t\in[0,T]; \quad i=1,2. \tag{2} \]
Lemma. Suppose there exists in \(\overline D\) a continuous solution of problem (1)—(2) with derivatives \(u_t,u_x,u_{xx}\) continuous in \(\overline D\), and that the conditions
\[ a(x,t)\geqslant a_0=\mathrm{const}>0 \quad \text{for } x,t\in\overline D; \tag{A} \]
\[ \{|u(x,t)|,\ a(x,t),\ |b(x,t)|,\ |c(x,t)|,\ |f(x,t)|\}\leqslant K_1 \quad \text{for } x,t\in\overline D; \tag{B} \]
\[ \{|\varphi_i(t)|,\ |\psi_i(t)|,\ |\varphi_i'(t)|,\ |\psi_i'(t)|\}\leqslant K_2 \quad \text{for } t\in[0,T]; \tag{C} \]
\[ |\varphi(x')-\varphi(x'')|\leqslant K_3|x'-x''|^\alpha \quad \text{for } x',x''\in[\varphi_1(0),\varphi_2(0)],\quad 0<\alpha<1; \tag{D} \]
\[ \varphi[\varphi_i(0)]=\psi_i(0) \quad (i=1,2), \tag{E} \]
are fulfilled, where \(K_i\) \((i=1,2,3)\) are nonnegative constants. Then the inequality
\[ |u(x',t')-u(x'',t'')| \leqslant K\bigl(|x'-x''|^\alpha+|t'-t''|^{\alpha/2}\bigr) \]
\[ \text{for } x',t';\ x'',t''\in\overline D; \tag{3} \]
holds; \(K\) is a nonnegative constant depending only on \(K_i,\alpha,a_0\), and the diameter of the trapezoid \(D\).
Proof. In the planes \((x,t)\), \((y,\tau)\) consider, respectively, the sets \(D_1,\overline D_1,\Gamma_1\) and \(D_2,\overline D_2,\Gamma_2\), analogous to \(D,\overline D,\Gamma\). In the space \((x,y,t,\tau)\) construct the direct product \(\overline D_3=\overline D_1\times\overline D_2\).
Let \(\Gamma_3=(\Gamma_1\times \overline{D}_2)\cup(\Gamma_2\times \overline{D}_1)\). If the point \((x,y,t,\tau)\in \overline{D}_3\setminus \Gamma_3\), then at it we may write
\[ L_x u(x,t)=0,\qquad L_y u(y,\tau)=0. \tag{4} \]
On \(\overline{D}_3\) consider the function
\[
v=v(x,y,t,\tau)=|u(x,t)-u(y,\tau)|^{2/\alpha}/[(x-y)^2+k|t-\tau|]
\]
\[
=|u_1-u_2|^{2/\alpha}/\rho\equiv |\Phi|^{2/\alpha}/\rho,
\]
where \(k\) is a certain positive constant, whose choice will be determined below. We shall seek \(\max v\) on \(\overline{D}_3\). Note that when \(x=y,\ t=\tau>0\), \(v=0\), and when \(x\ne y,\ t=\tau=0\), \(v\le k_3^{2/\alpha}\). Let \(v\) attain its greatest value at a point \(A(x,y,t,\tau)\), with \(t\ne \tau\). Then either \(A\in \overline{D}_3\setminus \Gamma_3\), or \(A\in \Gamma_3\). Suppose, for example, that \(A\in \overline{D}_3\setminus \Gamma_3\). Then at it (4) holds, and also
\[ v_x=0,\qquad v_y=0,\qquad v_{xx}\le 0,\qquad v_t\ge 0,\qquad v_\tau\ge 0,\qquad v_{yy}\le 0, \tag{5} \]
whence it follows that
\[ 2a(x,t)v_{xx}+a(y,\tau)v_{yy}-(2v_t+v_\tau)\le 0. \tag{6} \]
In more detail:
\[ v_x=2|\Phi|^{2/\alpha-1}u_{1x}\delta/\alpha\rho-2(x-y)|\Phi|^{2/\alpha}/\rho^2=0, \tag{7} \]
where \(\delta=\operatorname{sgn}\Phi\),
\[
v_{xx}=2|\Phi|^{2/\alpha-1}u_{1xx}\delta/\alpha\rho
+2(2-\alpha)|\Phi|^{2/\alpha-2}u_{1x}^2/\alpha^2\rho
\]
\[
-8(x-y)|\Phi|^{2/\alpha-1}u_{1x}\delta/\alpha\rho^2
+8(x-y)^2|\Phi|^{2/\alpha}/\rho^3
-2|\Phi|^{2/\alpha}/\rho^2\le 0.
\]
By virtue of (7) and \((x-y)^2/\rho<1\), we may write
\[ 0\ge v_{xx}\ge 2|\Phi|^{2/\alpha-1}u_{1xx}\delta/\alpha\rho-\overline{k}|\Phi|^{2/\alpha}/\rho^2,\qquad 0<\overline{k}<6. \tag{8} \]
Similarly,
\[ 0\ge v_{yy}\ge -2|\Phi|^{2/\alpha-1}u_{2yy}\delta/\alpha\rho-\overline{k}|\Phi|^{2/\alpha}/\rho^2. \tag{9} \]
Further:
\[ 2v_t+v_\tau =2|\Phi|^{2/\alpha-1}[2u_{1t}-u_{2\tau}]\delta/\alpha\rho -k|\Phi|^{r/\alpha}\rho^2,\qquad t>\tau. \]
Combining (6) and applying (4), with the help of (8) and (9) we obtain
\[
\{k-\overline{k}[a(x,t)+a(y,\tau)]\}|\Phi|^{2/\alpha}/\rho+
\]
\[
+\frac{2}{\alpha}|\Phi|^{2/\alpha-1}\delta\bigl[b(x,t)u_{1x}+c(x,t)u_1+f(x,t)-b(y,\tau)u_{2y}-c(y,\tau)u_2-f(y,\tau)\bigr]\le 0.
\]
Expressing \(u_{1x}\) and \(u_{2y}\) from the equalities \(v_x=v_y=0\), we obtain the inequality
\[ \{k-\overline{k}[a(x,t)+a(y,\tau)]\}|\Phi|^{2/\alpha}/\rho \le \overline{M}|\Phi|^{2/\alpha}/\rho+\overline{N}, \]
where \(\overline{M}\) and \(\overline{N}\) are easily computable constants depending only on \(K_1,\alpha\), and the diameter of \(D\). Choosing now \(k\) sufficiently large, we obtain the desired estimate.
In the remaining possible positions of the point \(A\), for which \(v\) attains its greatest value in \(\overline{D}_3\), the function \(v\) is investigated in an analogous way. The lemma is proved.
A function \(F(x,t,u)\), defined on the set \(G:\ x,t\in \overline{D},\ |u|\le M\), will be said to belong to the class \(C_{\gamma,\gamma/2,\lambda}(G)\) \((0<\gamma,\lambda<1)\), if for any points \((x',t',u')\), \((x'',t'',u'')\) from \(G\) the inequality
\[ |F(x',t',u')-F(x'',t'',u'')|\le K\bigl(|x'-x''|^\gamma+|t'-t''|^{\gamma/2}+|u'-u''|^\lambda\bigr), \]
holds, where \(K=\operatorname{const}\ge 0\).
Let the function \(g(x,t)\) be defined on \(\overline D\). Introduce the following notation \((^5)\):
\[ H[g]=\sup |g(x',t')-g(x'',t'')|/(|x'-x''|^\gamma+|t'-t''|^{\gamma/2}) \]
for \(x',t';\,x'',t''\in \overline D\);
\[ |g|_\gamma^D=\sup_{\overline D}|g|+H[g],\qquad |g|_{2+\gamma}^D=\sup_{\overline D}\left(\sum_{i=0}^2 |D_x^i g|\right)+\sum_{i=0}^2 H[D_x^i g]+|D_t g|_\gamma^D . \]
Theorem 1. Suppose that the following conditions are satisfied:
-
\(a(x,t,u)\ge 0,\ \partial f/\partial u\ge \beta=\mathrm{const}>0\) for \(x,t\in\overline D,\ |u|<\infty\).
-
\(a(x,t,u)\ge a_0=\mathrm{const}>0,\ \{a,|b|,|f|\}\le K_1;\ a,b,\partial f/\partial u\in C_{\gamma,\gamma/2,\lambda[G]}\),
where \(G:\ x,t\in\overline D,\ |u|\le M=\max_{\overline D}|f(x,t,0)|/\beta;\ K_1=\mathrm{const}\ge 0\).
-
The functions \(\varphi_i(t)\) \((i=1,2)\), defining the contour, are such that
\[ |\varphi_i(t_1)-\varphi_i(t_2)|\le K_2|t_1-t_2|^\delta \]
for \(t_1,t_2\in[0,T]\), \(\delta=\frac12\min(\gamma,\alpha\lambda)\), and \(K_2=\mathrm{const}\ge0\), \(0<\alpha<1\). -
\(f[\varphi_1(0),0,0]=f[\varphi_2(0),0,0]=0\).
Then the problem
\[ Lu\equiv u_t-a(x,t,u)u_{xx}+b(x,t,u)u_x+f(x,t,u)=0 \quad \text{for } x,t\in\overline D\setminus\Gamma; \tag{1'} \]
\[ u|_\Gamma=0 \tag{2'} \]
has at least one solution, and moreover
\[
|u|_{2+\delta}^D\le K_3|f(x,t,0)|_\gamma^D,
\]
where the constant \(K_3\) depends only on \(K,K_1,K_2,M,a_0,\delta\), and the diameter of \(D\).
Proof. Any possible solution of problem \((1')\)—\((2')\) of the smoothness required in this theorem is such that \(\max_{\overline D}|u|\le M\) (this follows from the maximum principle), and such that \(|u|_\alpha^D\le K_4\) (this follows from our lemma), where the constant \(K_4\) depends only on \(K_1,K_2,\alpha,a_0,M\), and the diameter of \(D\).
On the domain \(\overline D\), consider the Banach space of functions \(v(x,t)\) such that \(|v|_\alpha^D<\infty\) and \(v|_\Gamma=0\). Denote this space by \(A\). In it take the ball \(|v|_\alpha^D\le M+K_4\). Define the transformation \(u=u(v)\) as the solution of the problem
\[ u_t-a(x,t,v)u_{xx}+b(x,t,v)u_x +u\int_0^v \frac{\partial f(x,t,y)}{\partial y}\,dy +f(x,t,0)=0; \tag{1''} \]
\[ u|_\Gamma=0. \tag{2''} \]
Problem \((1)\)—\((2)\) has a solution \((^5)\). Concerning the transformation \(u=u(v)\), one can say: 1) \(|u|_\alpha^D\le M+K_4\); 2) \(|u|_{2+\delta}^D\le K_3|f(x,t,0)|_\gamma^D\) \((^5)\). Hence it is clear that the hypotheses of Schauder’s theorem \((^6)\) on a fixed point of a continuous, completely continuous transformation of the ball in \(A\) into itself are satisfied. The theorem is proved.
Theorem 2. Suppose that the following conditions are satisfied:
-
\(a(x,t,u)\ge 0;\ h(x,t,u),\partial f/\partial u\) are continuous for \(x,t\in\overline D,\ |u|<\infty\), and \(\partial f/\partial u\ge\beta=\mathrm{const}>0\) in the indicated domain.
-
\(a(x,t,u)\ge a_0=\mathrm{const}>0\) for \(x,t\in\overline D,\ |u|\le M\) (see Theorem 1).
-
\(a,b,f,h,a_u,b_u,f_u,h_u,a_x,h_x,a_t,h_t,a_{xx},h_{xx}\in C_{\gamma,\gamma/2,\lambda[G]}\).
-
\(|\varphi_i'(t_1)-\varphi_i'(t_2)|\le K_1|t_1-t_2|^{\gamma/2},\ (t_1,t_2)\in[0,T],\ i=1,2\).
-
\(f[\varphi_1(0),0,0]=f[\varphi_2(0),0,0]=0\).
Then there exists a unique solution of the problem
\[ u_t-a(x,t,u)u_{xx}+b(x,t,u)u_x+h(x,t,u)u_x^2+f(x,t,u)=0,\quad x,t\in D\setminus\Gamma; \]
\[ u|_\Gamma=0. \]
such that \(|u|_{2+\alpha}^{D} \leq K |f(x,t,0)|_{\gamma}^{D}\), \(0<\alpha<1\).
Proof. By the substitution \(v=\displaystyle\int_0^u e^{c(p)}\,dp\), where \(|u|\leq M\),
\[ c(p)=\int_0^p \frac{h(x,t,l)}{a(x,t,l)}\,dl, \]
this problem is essentially reduced to a problem of type \((1')\)—\((2')\).
The following two theorems are proved with the aid of the existence theorems given in \((^{1,3})\) and the estimates given in \((^{2,4,5})\).
Theorem 3. Suppose the following conditions are satisfied:
-
\(a\geq a_0,\quad f_u\leq -\beta\) \((a_0,\beta=\mathrm{const}>0)\) for \(x,t\in \overline{D}=[0,1]\times[0,T]\), \(|u|,\ |u_x|<\infty\).
-
In the domain \(|u|\leq M=\max_{\overline D}|f(x,t,0,0)|/\beta,\ |u_x|<\infty\), there exist continuous \(a_x, f_x, a_{u_x}, f_{u_x}\), and moreover \(|f|/a\leq K_1u_x^2+K_2u\); for sufficiently large \(|u_x|\) the inequality \(|f_x|/|f_u|\leq K_3 |u_x|^{1-\varepsilon}+K_4\) holds, where \(\varepsilon>0\), \(K_i\) \((i=1,2,3,4)\) are nonnegative constants.
-
\(a, f, a_x, f_x, a_t, f_t, a_{u_x}, f_{u_x}, f_u\) satisfy, with respect to the arguments, a Hölder condition with exponent \(0<\gamma<1\) and Hölder constants less than \(K_5\) for \(x,t\in\overline D,\ |u|\leq M,\ |u_x|\leq M_1\).
-
\(f(0,0,0,0)=f(1,0,0,0)=0\).
Then there exists a unique solution of the problem
\[ u_t=a(x,t,u_x)u_{xx}+f(x,t,u,u_x),\qquad xt\in \overline D\setminus\Gamma; \]
\[ u|_{\Gamma}=0, \]
and, moreover, there is a \(0<\alpha<1\) such that \(|u|_{2+\alpha}^{D}\leq K_6=\mathrm{const}\geq0\).
Theorem 4. Suppose the following conditions are satisfied:
-
\(a\geq0,\quad f_u\leq-\beta\) for \(x,t\in\overline D,\ |u|,\ |u_x|<\infty,\ \beta=\mathrm{const}>0\).
-
\(a\geq a_0=\mathrm{const}>0,\ |f|/a\leq K_1u_x^2+K_2u\); for \(|u_x|\gg0\) the inequality \(|f_x|/|f_u|\leq K_3|u_x|^{1-\varepsilon}+K_4\) holds for \(x,t\in\overline D,\ |u|\leq M,\ |u_x|<\infty,\ \varepsilon>0\).
-
\(a_x, f_x, a_u, f_u, a_{u_x}, f_{u_x}\) are continuous with respect to the arguments, for \(x,t\in\overline D,\ |u|\leq M,\ |u_x|<\infty\).
-
All partial derivatives of \(a, f\) with respect to \(x,u,u_x\) up to and including second order satisfy, with respect to the arguments, a Hölder condition with exponent \(0<\gamma<1\) for \(x,t\in\overline D,\ |u|\leq M,\ |u_x|\leq M_1\).
-
\(2|a_{u_x}|/a<K_0,\ |a_{u_xu_x}|/a<K_0\), where \(K_0=\varepsilon/G(M_1\sqrt{\rho}+3M_1^2)\) \((^2)\), for \(x,t\in\overline D,\ |u|\leq M,\ |u_x|\leq M_1\).
-
\(f(0,0,0,0)=f(1,0,0,0)=0\).
Then there exists a unique solution of the problem
\[ u_t=a(x,t,u,u_x)u_{xx}+f(x,t,u,u_x),\qquad x,t\in \overline D\setminus\Gamma. \]
\[ u|_{\Gamma}=0, \]
and, moreover, there is a \(0<\alpha<1\) such that \(|u|_{2+\alpha}^{D}\leq K_5=\mathrm{const}\geq0\).
In conclusion, the author expresses gratitude to O. A. Ladyzhenskaya for posing the problem and for discussing the work.
Leningrad State University
named after A. A. Zhdanov
Received
1 IV 1960
CITED LITERATURE
\({}^{1}\) F. Browder, Duke Math. J., 24, 579 (1957).
\({}^{2}\) O. A. Ladyzhenskaya, Tr. Moskovsk. matem. obshch., 7, 150 (1958).
\({}^{3}\) C. Miranda, Equations with partial derivatives of elliptic type, Moscow, 1957.
\({}^{4}\) A. Friedman, J. Math. and Mech., 7, No. 3 (1958).
\({}^{5}\) A. Friedman, J. Math. and Mech., 7, No. 5 (1958).
\({}^{6}\) J. Schauder, Studia Math., 2, Fasc. IV, 171 (1930).