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R. Ya. Glagoleva
CONTINUOUS DEPENDENCE ON THE INITIAL DATA OF THE SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM FOR A PARABOLIC EQUATION WITH NEGATIVE TIME
(Presented by Academician I. G. Petrovskii, 30 VI 1962)
In this note we consider the parabolic equation
\[ \frac{\partial u}{\partial t} = \sum_{i,k=1}^{n} \frac{\partial}{\partial x_i} \left[ a_{ik}(x_1,\ldots,x_n)\frac{\partial u}{\partial x_k} \right] + c(x_1,\ldots,x_n)u + f(t,x_1,\ldots,x_n). \tag{1} \]
Let \(R\) be a cylinder in the \((n+1)\)-dimensional space \((t,x_1,\ldots,x_n)\), with generators parallel to the \(t\)-axis, whose upper and lower bases lie respectively in the hyperplanes \(t=0\) and \(t=-T\). Let \(\Gamma\) be the lateral surface of the cylinder and \(D\) the domain of the \(n\)-dimensional space lying in its base. The boundary of \(D\) is assumed smooth. The following problem is considered:
\[ u\big|_{t=0}=\varphi(x,\ldots,x_n),\qquad u\big|_{\Gamma}=\psi(t,x_1,\ldots,x_n). \]
It is known that such a problem is ill posed: a solution does not exist for every initial function \(\varphi(x_1,\ldots,x_n)\), and there is no continuous dependence on the initial data. However, if a solution exists, then it is unique \((^{1,2})\). In this note it will be shown that, in the class of solutions uniformly bounded in \(R\), there is continuous dependence on the initial function, and an estimate will be given for the change in the solution as a function of the change in the initial function and of the constant bounding the solutions.
It is assumed that the coefficients in (1) satisfy the conditions
\[ a_{ik}=a_{ki};\qquad \lambda_2^2\sum_{i=1}^{n}\xi_i^2 \ge \sum_{i,k=1}^{n}a_{ik}\xi_i\xi_k \ge \lambda_1^2\sum_{i=1}^{n}\xi_i^2;\qquad -K<c\le 0. \tag{2} \]
In the first part we shall assume that the coefficients are only measurable and, under this assumption, obtain an estimate in the \(L_2\) metric. Here \(u\) is a generalized solution—continuous in the aggregate of the variables, possesses generalized Sobolev derivatives with respect to \(x_i\) from \(L_2\), and satisfies the integral identity (\(\varphi\) is an arbitrary continuously differentiable function vanishing on the boundary of \(R\)):
\[ \iint_R \left[ u\frac{\partial\varphi}{\partial t} - \sum_{i,k=1}^{n} a_{ik}\frac{\partial u}{\partial x_i}\frac{\partial\varphi}{\partial x_k} + cu\varphi \right]\,dX\,dt =0. \tag{3} \]
In the second part continuous differentiability of the coefficients \(a_{ik}\) will be assumed, and the estimate will then be obtained in the \(C\) metric.
1°. Introduce the following notation. For every function \(f(t,x_1,\ldots,x_n)\) in the cylinder \(R\), set
\[ \|f\|_t=\left(\int_D f^2(t,x_1,\ldots,x_n)\,dX\right)^{1/2}. \]
Theorem 1. Let \(u_1\) and \(u_2\) be two solutions of equation (1) in \(R\) such that they coincide on \(\Gamma\) and
\[ \|u_1-u_2\|_0<\Delta;\qquad \|u_1-u_2\|_{-T}<M. \tag{4} \]
Then for every \(t\), \(0>t>-T/40\), and for sufficiently small \(\Delta\), the inequality
\[ \|u_1-u_2\|_t<\frac{Q}{\sqrt{T}}\Delta^{a/\ln(0,1T)}, \tag{5} \]
holds, where \(a<0\) is an absolute constant and \(Q\) is a constant depending on \(M, K, \lambda_1, \lambda_2\) and on the diameter of the domain \(D\).
If we prove the theorem under the assumption that the coefficients are smooth, then the required estimate for nonsmooth coefficients can be obtained by passage to the limit. We shall therefore assume henceforth that the coefficients are continuously differentiable and that the solution is classical. We precede the proof of the theorem by two lemmas.
Lemma 1. Consider the equation
\[ \frac{\partial u}{\partial t} +\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(a_{ik}\frac{\partial u}{\partial x_k}\right)+cu=0, \tag{1'} \]
where the coefficients satisfy the conditions listed above. Let \(u\) be a solution of equation \((1')\) in the cylinder
\(R'=\{0<t_0<t<T<1,\ X\in D\}\), vanishing on its lateral surface \(\Gamma'\). Let
\[ \|u\|_{t_0}<t_0^\beta;\qquad \|u\|_T<M;\qquad \left\|\left[\sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_i}\right)^2\right]^{1/2}\right\|_T<M, \tag{6} \]
where \(\beta>1,\ M>0\) are arbitrary numbers. Then for any \(t,\ t_0<t<T/2\), the inequality
\[ \|u\|_t<A(4t/T)^{\beta-1/2}, \tag{7} \]
holds, where \(A\) is a constant depending on \(K, \lambda_1, \lambda_2, M\) and the diameter of the domain \(D\).
Proof. Introduce the auxiliary function \(v=u/t^\beta\). The function \(v\) satisfies the equation
\[ \beta v+t\frac{\partial v}{\partial t} +t\sum_{i,k=1}^{n}\frac{\partial}{\partial x_i} \left(a_{ik}\frac{\partial v}{\partial x_k}\right)+ctv=0. \]
Multiplying by \(\partial v/\partial t\) and integrating over \(R'\), we find
\[ \frac{\beta}{2}\int_{R'}\frac{\partial v^2}{\partial t}\,dX\,dt +\int_{R'}t\left(\frac{\partial v}{\partial t}\right)^2dX\,dt - \]
\[ -\int_{R'}t\left[ \sum_{i,k=1}^{n}a_{ik}\frac{\partial v}{\partial x_i} \frac{\partial^2 v}{\partial x_k\partial t} \right]dX\,dt +\int_{R'}\frac{ct}{2}\frac{\partial v^2}{\partial t}\,dX\,dt=0. \]
Taking into account the condition \(a_{ik}=a_{ki}\) and the zero boundary conditions, we obtain
\[ -\int_{R'}t\left[ \sum_{i,k=1}^{n}a_{ik}\frac{\partial v}{\partial x_i} \frac{\partial^2 v}{\partial x_k\partial t} \right]dX\,dt = -\frac{T}{2}\int_D \left[\sum_{i,k=1}^{n}a_{ik}\frac{\partial v}{\partial x_i} \frac{\partial v}{\partial x_k}\right]_{t=T}dX + \]
\[ +\frac{t_0}{2}\int_{D_0} \left[\sum_{i,k=1}^{n}a_{ik}\frac{\partial v}{\partial x_i} \frac{\partial v}{\partial x_k}\right]_{t=t_0}dX +\frac{1}{2}\int_{R'} \left[\sum_{i,k=1}^{n}a_{ik}\frac{\partial v}{\partial x_i} \frac{\partial v}{\partial x_k}\right]dX\,dt, \]
and straightforward calculations lead us to the inequality
\[ \int_{R'}\left[\sum_{i=1}^{n}\left(\frac{\partial v}{\partial x_i}\right)^2\right]dX\,dt < \frac{1}{2\lambda_1^2} \left[ \beta+\frac{M^2(\lambda_2^2+K)}{T^{2\beta-1}} \right] \]
or
\[ \int_{R'}v^2\,dX\,dt < \frac{1}{2G\lambda_1^2} \left[ \beta+\frac{M^2(\lambda_2^2+K)}{T^{2\beta-1}} \right], \]
where \(G\) is a constant depending on the diameter of the domain \(D\). Passing to \(u\) and noting that \(\beta < (2/T)^{2\beta-1}\), we find
\[ \int_{R'} \frac{u^{2}}{t^{2\beta}}\,dX\,dt < \frac{1+M^{2}(\lambda_{2}^{2}+K)}{2G\lambda_{1}^{2}} \left(\frac{2}{T}\right)^{2\beta-1}. \tag{8} \]
Let now \(t_{1}\) be an arbitrary number satisfying the inequality \(t_{0}<t_{1}<T/2\). Then from (8) we find
\[ \int_{t_{1}}^{2t_{1}}\int_{D} u^{2}\,dX\,dt < \frac{1+M^{2}(\lambda_{2}^{2}+K)}{G\lambda_{1}^{2}} \left(\frac{4t_{1}}{T}\right)^{2\beta-1} t_{1}. \]
Applying the mean-value theorem to the outer integral and taking into account that \(d(\|u\|_{t}^{2})/dt\geqslant 0\), we obtain for all \(t,\ t_{0}<t\leqslant T/2\),
\[ \|u\|_{t}<A(4t/T)^{\beta-\frac12}, \]
where \(A=\sqrt{[1+M^{2}(\lambda_{2}^{2}+K)]/G\lambda_{1}^{2}}\), which was to be proved.
Lemma 2. Suppose that for equation \((1')\) in the cylinder \(R'\) we have a solution \(u\) which vanishes on the lateral surface \(\Gamma'\), and
\[ \|u\|_{t_{0}}<t_{0}^{\beta}, \qquad \|u\|_{T}<M_{1}. \tag{9} \]
Then for all \(t,\ t_{0}<t\leqslant T/4\),
\[ \|u\|_{t}<\frac{A}{\sqrt{T}}\left(\frac{8t}{T}\right)^{\beta-\frac12}. \tag{10} \]
Proof. From equation (1) we find
\[ \int_{T/2}^{T} \left\| \left[ \sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_{i}}\right)^{2} \right]^{1/2} \right\|_{t}^{2}\,dt < \frac{M_{1}^{2}}{2\lambda_{1}^{2}}, \]
and by the mean-value theorem there exists \(T_{0}\), \(T/2<T_{0}<T\), such that
\[ \left\| \left[ \sum_{i=1}^{n}\left(\frac{\partial u}{\partial x_{i}}\right)^{2} \right]^{1/2} \right\|_{T_{0}} < \frac{M_{1}}{\lambda_{1}\sqrt{T}}. \]
Put \(u_{1}=u\sqrt{T}\). Then, by condition (9) and taking into account that \(T<1\), we obtain
\[ \|u_{1}\|_{T_{0}}<\|u_{1}\|_{T}<M_{1}\sqrt{T}<M_{1}, \qquad \left\| \left[ \sum_{i=1}^{n}\left(\frac{\partial u_{1}}{\partial x_{i}}\right)^{2} \right]^{1/2} \right\|<\frac{M_{1}}{\lambda_{1}}. \]
Put \(M=\max\{M_{1},\,M_{1}/\lambda_{1}\}\) and, applying Lemma 1 to \(u_{1}\) in the cylinder \(R_{0}'=\{t_{0}<t<T_{0},\ x\in D\}\), we obtain for \(t,\ t_{0}<t\leqslant T_{0}/2\), by inequality (7),
\[ \|u_{1}\|_{t}<A_{1}(4t/T_{0})^{\beta-\frac12}. \]
Passing from \(u_{1}\) to \(u\) and noting that \(T_{0}>T/2\), we obtain for \(t,\ t_{0}<t\leqslant T/4\),
\[ \|u\|_{t}<\frac{A}{\sqrt{T}}\left(\frac{8t}{T}\right)^{\beta-\frac12}, \]
which was to be proved.
Proof of the theorem. Put \(U=u_{1}-u_{2}\). \(U\) satisfies the homogeneous equation
\[ \frac{\partial U}{\partial t} = \sum_{i,k=1}^{n} \frac{\partial}{\partial x_{i}} \left(a_{ik}\frac{\partial U}{\partial x_{k}}\right) +cU \tag{11} \]
and the conditions
\[ \|U\|_{0}<\Delta;\qquad \|U\|_{-T}<M. \tag{12} \]
Extend \(U\) to positive values of \(t\), solving the first boundary-value problem with zero boundary conditions (which is possible, since in this case the problem is posed correctly). Make the change of variable \(t\), setting \(t=t_0-\tau\) \((t_0>0)\) and \(V(\tau,x)=U(t_0-\tau,x)\). Then equation (11) will pass into \((1')\) with respect to the function \(V\), and conditions (12) into the conditions
\[ \|V\|_{t_0}<\Delta,\qquad \|V\|_{T+t_0}<M. \]
Represent \(\Delta\) in the form \(\Delta=t_0^{\beta}\) \((\beta=\ln\Delta/\ln t_0)\). We shall assume that \(\Delta<t_0\). From Lemma 2 it follows that, for every \(\tau\), \(t_0<\tau<(T+t_0)/4\), the inequality
\[ \|V\|_{\tau}<\frac{A}{\sqrt{8\tau}}\, \Delta^{\ln\left(\frac{8\tau}{T+t_0}\right)/\ln t_0}. \]
holds. Let \(t_0=0.1\,T\) and \(t_0<\tau<T/8\). Then
\[ \|V\|_{\tau}<\frac{Q}{\sqrt{T}}\, \Delta^{-\ln 1.1/\ln 0.1T},\qquad \text{where } Q=\frac{A}{\sqrt{0.8}}. \tag{13} \]
Returning to \(U\), we obtain from (13), for \(t\), \(-T/40\le t<0\),
\[ \|U\|_t<\frac{Q}{\sqrt{T}}\, \Delta^{-\ln 1.1/\ln(0.1\,T)} \]
and, putting \(-\ln 1.1=a\), we obtain the assertion of the theorem.
\(2^\circ\). Let the coefficients \(a_{ik}\) be continuously differentiable and let their derivatives be bounded by the constant \(K\).
Theorem 2. Let \(u_1\) and \(u_2\) be two solutions of equation (1) in \(R\) such that they coincide on \(\Gamma\) and
\[ |u_1-u_2|_{t=0}<\Delta,\qquad |u_1-u_2|_{t=-T}<M. \]
Then, for every \(t\), \(0>t\ge -T/40\), and for sufficiently small \(\Delta\), the inequality
\[ |u_1-u_2|<\frac{Q_1}{T^{(n+1)/2}}\, \Delta^{a/\ln 0.1T}, \]
holds, where \(a<0\) is an absolute constant and \(Q_1\) is a constant depending on \(M, K, \lambda_1, \lambda_2\), and the diameter of the domain \(D\).
In the proof one uses Pogorzelski’s theorem \((^3)\).
\(3^\circ\). In Theorems 1 and 2 it was assumed that \(T\ll 1\), and the estimates were obtained in the part of the cylinder adjacent to the hyperplane \(t=0\) \((0>t\ge -T/40)\).
For a cylinder of arbitrary size, and for any point in it, estimates can be obtained by partitioning it into parts and successively applying Theorem 1 or Theorem 2 to each part.
\(4^\circ\). Analogous results are obtained for the equation
\[ u_t=\sum a_{ik}(t,x)u_{x_i x_k}+\sum b_i(t,x)u_{x_i}+c(t,x)u+f(t,x), \]
if, in addition, one assumes continuous differentiability with respect to \(t\) of the coefficients \(a_{ik}\).
Moscow Aviation Institute
named after S. Ordzhonikidze
Received
26 VI 1962
CITED LITERATURE
- Ito, H. Jamable, J. Math. Soc. Japan, 10, No. 3 (1958).
- J. L. Lions, B. Malgrange, Math. Scand., 8, No. 2, 277 (1960).
- V. Pogorzelski, Bull. Polish Acad. Sci., ser. III, 4, 1, 9 (1956).