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MATHEMATICS
E. K. ISAKOVA
ON THE CAUCHY PROBLEM FOR PARABOLIC EQUATIONS WITH A SMALL PARAMETER
(Presented by Academician S. L. Sobolev, VII 5, 1962)
Consider the solution \(u^\varepsilon(x,t)\) of the \(2p\)-parabolic \({}^{(1)}\) equation
\[ \widetilde L_\varepsilon u^\varepsilon(x,t) \equiv L_\varepsilon^1 u^\varepsilon(x,t) + L_\varepsilon^2 u^\varepsilon(x,t) + L_0 u^\varepsilon(x,t)=0, \tag{1} \]
satisfying the condition
\[ \left. u^\varepsilon(x,t)\right|_{t=0} = \Psi(x)\equiv \psi(x)\chi(x_1), \tag{2} \]
where
\[ L_\varepsilon^1 u^\varepsilon = (-1)^p \sum_{s\le |i|\le 2p} a_i(x,t)\varepsilon^{|i|/2p}D^i u^\varepsilon; \]
\[ L_\varepsilon^2 u^\varepsilon \equiv \varepsilon^{s/2p} \sum_{1\le |i|\le s-1} a_i(x,t)D^{|i|}u^\varepsilon; \]
\[ L_0u^\varepsilon(x,t) =\sum b_i(x,t)\frac{\partial u^\varepsilon}{\partial x_i} +\frac{\partial u^\varepsilon}{\partial t} +c(x,t)u^\varepsilon(x,t); \]
\[ \varepsilon>0;\quad s\text{ is an integer},\ 1\le s\le 2p-1;\quad i=(i_1,\ldots,i_n),\quad |i|=i_1+\cdots+i_n; \]
\[ x=(x_1,\ldots,x_n)\in E^n\ (-\infty<x_i<\infty);\quad D^i=\partial^{|i|}/\partial x_1^{i_1}\cdots\partial x_n^{i_n},\quad t\in[0,T]; \]
\(\chi(x_1)\) is equal to zero for \(x_1<0\) and to one for \(x_1\ge 0\). For simplicity of exposition we shall assume that all coefficients in (1) and the function \(\psi(x)\) belong to \(C^\infty(E^n\times[0,T])\) and are bounded in \(E^n\times[0,T]\) together with all their derivatives.
We shall be interested in the behavior of the solution \(u^\varepsilon(x,t)\) of problem (1)—(2) as \(\varepsilon\to0\). In the case \(p=1\) this problem was considered in \({}^{(2,3)}\). Denote by \(u^0(x,t)\) the solution of problem (1)—(2) for \(\varepsilon=0\) (we shall call it problem \((1^0)—(2^0)\)), and by \(l(x,t)\) the characteristic of equation \((1^0)\) passing through the point \((x,t)\). The solution \(u^0(x,t)\) has a discontinuity of the first kind along the characteristics \(l(x,t)\) passing through the plane \(x_1=t=0\). These characteristics will be called discontinuity characteristics, and the surface which they form will be called the discontinuity surface.
Theorem 1. As \(\varepsilon\to0\), the solution \(u^\varepsilon(x,t)\) of problem (1)—(2) converges to the solution \(u^0(x,t)\) of problem \((1^0)—(2^0)\) everywhere outside the discontinuity surface of the function \(u^0(x,t)\). This convergence is uniform in \((x,t)\in E^n\times[0,T]\) outside any \(\delta\)-neighborhood of the discontinuity surface of the function \(u^0(x,t)\), \(\delta>0\).
In what follows we shall assume that the operator \(L_0\) has the form \(L_0u=\partial u/\partial t+c(x,t)u\) (i.e., in (1) \(b_i(x,t)=0\)). This can be achieved by replacing \((x,t)\) by \((\bar x,\bar t)\), where \(\bar t\) is the length of the arc of the characteristic \(l(x,t)\) between the points \((\bar x,0)\) and \((x,t)\), and \(\bar x=(\bar x_1,\ldots,\bar x_n)\) are the coordinates of the point of intersection of \(l(x,t)\) with the plane \(t=0\).
I. Let \(n=1\). Then problem (1)—(2) takes the form:
\[ \widetilde L_{\varepsilon}(x,t)u^\varepsilon(x,t)\equiv \varepsilon a_{2p}(x,t)\frac{\partial^{2p}u^\varepsilon}{\partial x^{2p}} +\cdots+\varepsilon^{s/2p}a_s(x,t)\frac{\partial^s u^\varepsilon}{\partial x^s}+ \]
\[ +\varepsilon^{s/2p}\left\{a_{s-1}(x,t)\frac{\partial^{s-1}u^\varepsilon}{\partial x^{s-1}} +\cdots+a_1(x,t)\frac{\partial u^\varepsilon}{\partial x}\right\} +\frac{\partial u^\varepsilon}{\partial t}+c(x,t)u^\varepsilon=0, \tag{3} \]
\[ u^\varepsilon(x,t)\big|_{t=0}=\psi(x)\chi(x), \tag{4} \]
where \(x\in E'(-\infty<x<\infty)\), \(t\in[0,T]\), \(a_{2p}(x,t)=(-1)^p\widetilde a_{2p}(x,t)\), \(\widetilde a_{2p}(x,t)\geqslant\beta\geqslant0\).
Put \(\widetilde L_\varepsilon(x,t)\equiv L_{1\varepsilon}(x,t)+L_0(x,t)\), where \(L_0(x,t)\equiv\partial/\partial t+c(x,t)\), \(L_{1\varepsilon}=L_\varepsilon^1+L_\varepsilon^2\), \(\alpha(x,t)=\exp\left\{-\int_0^t c(x,\tau)\,d\tau\right\}\), and denote by \(U_0^\varepsilon(x,\xi,t,\tau)\) the fundamental solution of equation (3) in the case when the coefficients do not depend on \(x\), \(c(x,t)=0\), \(L_\varepsilon^2=0\). Consider the functions
\[ v_{kj}^\varepsilon\bigl(x,t,\Phi_\varepsilon(0,t)\bigr)\equiv v_{kj}^\varepsilon(\Phi_\varepsilon) =\varepsilon^{j/2p}\int_0^t a_j\frac{\partial^k\Phi_\varepsilon(0,t)}{\partial x^k} \frac{\partial^{j-1-k}U_0^\varepsilon(x,0,t,\tau)}{\partial x^{j-1-k}}\,d\tau \]
\[ (k=0,1,\ldots,j-1;\ j=1,\ldots,2p), \]
where \(\Phi_\varepsilon(x,t)\) is a function infinitely differentiable with respect to \(x,t\) and uniformly bounded in \(\varepsilon\), together with all its derivatives with respect to \(x,t\), for \((x,t)\in E'\times[0,T]\).
Lemma. If for all \((x,t)\in E'\times[0,T]\)
\[ |\Phi_\varepsilon(x,t)|\leqslant C\varepsilon^{l/2p}, \]
where the constant \(C>0\), then
\[ |v_{kj}^\varepsilon(\Phi_\varepsilon)| \leqslant C_1t\,(\varepsilon t)^{k/2p}\varepsilon^{l/2p} \quad \left(\text{i.e. }v_{kj}^\varepsilon=O\left(\varepsilon^{(l+k)/2p}\right)\right) \tag{5} \]
\[ (k=0,1,\ldots,j-1;\ j=1,\ldots,2p) \]
for \((x,t)\in E'\times[0,T]\), with a constant \(C_1>0\).
The proof of this lemma is based on estimates of the fundamental solution \(U_0^\varepsilon(x,\xi,t,\tau)\) obtained in \((4)\).
The functions \(v_{kj}^\varepsilon(\Phi_\varepsilon)\) defined above \((k=0,1,\ldots,j-1;\ j=1,\ldots,2p)\), for which the relations (5) are valid, will be called a \(2p\)-parabolic boundary layer of order \((k+l)\).
Theorem 2. If the coefficients of equation (3) do not depend on \(x\), \(L_\varepsilon^2=0\), then the solution \(u^\varepsilon(x,t)\) of problem (3)—(4) can be represented in the form
\[ u^\varepsilon(x,t)=u^0(x,t)+ \]
\[ +\alpha(t)\left\{ \sum_{i=0}^{l}\sum_{j=s}^{2p}\sum_{k=0}^{j-1}v_{kj}^\varepsilon(z^{i\varepsilon}) +\sum_{i=1}^{l}\chi(x)z^{i\varepsilon}(x,t) +\bar z^{(l+1)\varepsilon}(x,t) \right\}, \tag{6} \]
where \(u^0(x,t)\) is the solution of problem (3)—(4) for \(\varepsilon=0\); \(z^{0\varepsilon}(x,t)=\psi(x)\); \(z^{i\varepsilon}(x,t)\), \(i=1,2,\ldots,l\), are determined recurrently by
\[ \frac{dz^{(i+1)\varepsilon}}{dt} =-L_{1\varepsilon}(t)z^{i\varepsilon}(x,t),\qquad z^{(i+1)\varepsilon}\big|_{t=0}=0,\qquad i=0,1,\ldots,l-1; \]
\[ \bigl[L_{1\varepsilon}(t)+\partial/\partial t\bigr]\bar z^{(l+1)\varepsilon}(x,t) =-\bigl[L_{1\varepsilon}(t)z^{l\varepsilon}(x,t)\bigr]\chi(x), \qquad \bar z^{(l+1)\varepsilon}\big|_{t=0}=0. \]
Moreover,
\[ z^{i\varepsilon}(x,t)=O(\varepsilon^{is/2p}),\qquad i=1,\ldots,l,\qquad \bar z^{(l+1)\varepsilon}(x,t)=O(\varepsilon^{(l+1)s/2p}) \]
for \((x,t)\in E'[0,T]\); \(v_{kj}^\varepsilon(z^{i\varepsilon})\) \((k=0,1,\ldots,j-1;\ j=1,\ldots,2p)\) are functions of a \(2p\)-parabolic boundary layer of order \((k+is)\).
If \(L_\varepsilon^2 \ne 0\), then instead of (6) the following representation will hold for \(u^\varepsilon(x,t)\):
\[ u^\varepsilon(x,t)=w_0^\varepsilon+w_1^\varepsilon+\cdots+w_m^\varepsilon+ O\left(\varepsilon^{\frac{m+1}{2p}}\right), \tag{6'} \]
where (6) is valid for \(w_0^\varepsilon\), while \(w_{k+1}^\varepsilon=O(\varepsilon^{k/2p})\), \(k=0,\ldots,m-1\), are determined recursively:
\[ (L_\varepsilon^1+L_0)w_{k+1}^\varepsilon=-L_\varepsilon^2 w_k^\varepsilon, \qquad w_{k+1}^\varepsilon\big|_{t=0}=0. \]
We note that from (6) the validity of Theorem 1 in the case under consideration follows immediately.
Theorem 3. If the coefficients of equation (3) depend on \(x\) and \(t\), then for the solution \(u^\varepsilon(x,t)\) of problem (3)—(4) the representation
\[ u^\varepsilon(x,t)=\alpha(x,t)\{\overline{u}^{\,0}(x,t)+\overline{w}_0^\varepsilon(x,t)+w_1^\varepsilon+\cdots+w_m^\varepsilon\} +O(\varepsilon^{(m+1)/2p}), \]
holds, where \(\alpha(x,t)\overline{u}^{\,0}(x,t)=u^0(x,t)\). Moreover, for \(u^0+\overline{w}_0^\varepsilon\) in an \(\varepsilon^{1/2p}\)-neighborhood of \(x=0\) the representation (6′) is valid, while \(w_k^\varepsilon(x,t)=O(\varepsilon^{k/2p})\), \(k=1,\ldots,m\), uniformly with respect to \((x,t)\in E' \times [0,T]\).
II. Let \(n>1\). Denote by \(\widetilde{L}_{1\varepsilon}(x,t)\) that part of the operator \(\widetilde{L}_\varepsilon(x,t)\) which contains differentiation only with respect to \(x_1\), and by \(u_1^\varepsilon(x,t)\) the solution of the equation \(\widetilde{L}_{1\varepsilon}u_1^\varepsilon(x,t)=0\) under condition (2) (here \(x_2,\ldots,x_n\) are regarded as parameters).
Theorem 4. For the solution \(u^\varepsilon(x,t)\) of problem (1)—(2) there is the representation
\[ u^\varepsilon(x,t)=u_1^\varepsilon(x,t)+u_2^\varepsilon(x,t)+\cdots+u_{m+1}^\varepsilon(x,t) +O(\varepsilon^{(m+1)/2p}), \tag{7} \]
where \(u_k^\varepsilon\), \(k=2,\ldots,m+1\), are determined recursively:
\[ \widetilde{L}_{1\varepsilon}u_{k+1}^\varepsilon(x,t) = -\widetilde{L}_\varepsilon u_k^\varepsilon(x,t), \qquad u_{k+1}^\varepsilon(x,t)\big|_{t=0}=0, \qquad k=1,\ldots,m; \]
moreover, Theorem 3 is valid for \(u_1^\varepsilon(x,t)\), and
\(u_k^\varepsilon(x,t)=O(\varepsilon^{(k-1)/2p})\), \(k=2,\ldots,m+1\).
Computing Center
Academy of Sciences of the USSR
Received
30 VI 1962
CITED LITERATURE
- I. G. Petrovskii, Bull. Moscow State Univ., Section A, 7 (1938).
- E. K. Isakova, Dokl. Akad. Nauk SSSR, 117, No. 6 (1957).
- E. K. Isakova, Dokl. Akad. Nauk SSSR, 119, No. 6 (1958).
- O. A. Ladyzhenskaya, Mat. sbornik, 27 (69), 175 (1950).