V. A. KONDRAT'EV

GENERAL BOUNDARY-VALUE PROBLEMS FOR PARABOLIC EQUATIONS IN A CLOSED DOMAIN

(Presented by Academician I. G. Petrovskii, 31 XII 1964)

We shall consider, in a bounded closed domain \(G\) of the space \((t, x_1, \ldots, x_n)\), the equation

\[ L(t,x,\partial/\partial t,\partial/\partial x)u=f, \tag{1} \]

where \(L\) is a certain polynomial in the operators \(\partial/\partial t\), \(\partial/\partial x\) with coefficients depending on \(t, x_1,\ldots,x_n\). The principal part of the operator \(L\) is defined as follows.

Let \(l\) be the maximal order of the derivatives with respect to \(t\) entering the left-hand side of equation (1); \(2pl\) the maximal order of the derivatives with respect to \(x\). We shall call the generalized order of the derivative \(\partial^{k_1+k_2}/\partial t^{k_1}\partial x^{k_2}\) the number \(2pk_1+k_2\). Let the highest generalized order of the derivatives entering \(L\) be \(2pl\). We shall call the principal part of the operator \(L\) the operator \(L_0(t,x,\partial/\partial t,\partial/\partial x)\), which is obtained from \(L\) by discarding terms whose generalized order is less than \(2pl\). The number \(2pl\) will be called the generalized order of equation (1). It is assumed that \(L_0(t,x,i\tau,i\sigma)\ne0\) if \(|\tau|+|\sigma|\ne0\) and \(\tau,\sigma\) are real.

The boundary \(\Gamma\) of the domain \(G\) is assumed to be an infinitely differentiable surface. We suppose that on \(\Gamma\) there exist only two points \((t^0,x^0)\) and \((t^1,x^1)\) at which the tangent plane is perpendicular to the \(t\)-axis. Let \(t^0=0\), \(x^0=0\). We assume that, in a neighborhood of \((0,0)\), the equation of \(\Gamma\) has the form

\[ t^q=\sum_{i_1+\cdots+i_n=2pq} a_{i_1\ldots i_n}x_1^{i_1}\cdots x_n^{i_n} +o\bigl(|x|^{2pq}\bigr),\qquad q\text{ an integer}. \tag{2} \]

On \(\Gamma\) there are prescribed \(pl\) boundary conditions of the form

\[ B_j(t,x,\partial/\partial t,\partial/\partial x)u=\varphi_j, \tag{3} \]

where \(B_j\) are differential operators of generalized order \(m_j\).

The coefficients of the operators \(L\), \(B_j\) are infinitely differentiable everywhere except at the special points. At the point \((0,0)\) we assume that the coefficient \(a_{k_1k_2}(t,x)\) of the derivative \(\partial^{k_1+k_2}/\partial t^{k_1}\partial x^{k_2}\) of the operator \(L\) expands into the asymptotic series

\[ a_{k_1k_2} = \sum_{m=2k_1p+k_2-2lp}^{m=\infty} t^{m/2p} a_{k_1k_2}^{(m)} \left(\frac{x}{t^{1/2p}}\right), \tag{4} \]

where

\[ a_{k_1k_2}^{(2k_1p+k_2-2lp)}\equiv 0 \quad\text{for } 2k_1p+k_2<2lp. \]

The coefficient \(b_{k_1k_2}^{\,j}\) at the derivative \(\partial^{k_1+k_2}/\partial t^{k_1}\partial x^{k_2}\) of the operator \(B_j\) is expanded in the asymptotic series

\[ b_{k_1k_2}^{(j)} = \sum_{m=2k_1p+k_2-m_j}^{\infty} t^{m/2p} b_{k_1k_2}^{jm}\left(xt^{-1/2p}\right) \tag{5} \]

and \(b_{k_1k_2}^{jm}=0\) if \(2k_1p+k_2<m_j;\quad m=2k_1p+k_2-m_j\).

The functions \(a_{k_1k_2}^{m}\), \(b_{k_1k_2}^{jm}\) are infinitely differentiable. The operators obtained from \(L_0\), \(B_{j0}\)—the principal parts of the operators \(L\), \(B_j\)—by replacing \(a_{k_1k_2}\), \(b_{k_1k_2}^{j}\) by \(a_{k_1k_2}^{0}\), \(b_{k_1k_2}^{j0}\), respectively, are denoted by \(L_0(0)\), \(B_{j0}(0)\).

Expansions analogous to (4), (5) are also required in a neighborhood of the second distinguished point of the boundary. At all non-distinguished points of the boundary the operators \(L\), \(B_j\) must satisfy a condition analogous to the Shapiro–Lopatinskii condition for elliptic problems. One more condition is imposed at the distinguished point.

It is required that the special problem

\[ L_0(0)u=f, \]

\[ B_{j0}(0)u=\varphi_j \tag{6} \]

satisfy a condition of Shapiro–Lopatinskii type on the entire surface \(\Gamma_0\):

\[ t^q = \sum_{i_1+\cdots+i_n=2pq} a_{i_1\ldots i_n}x_1^{i_1}\cdots x_n^{i_n}, \tag{7} \]

except for the point \((0,0)\).

The study of problem (1), (3) begins with the investigation of problem (6) in the domain \(G_0\), bounded by the paraboloid (7). Define the space \(\dot H_{\alpha}^{k\,2p}(G_0)\), for integer \(k\), as the space with norm

\[ \|u\|_{\dot H_{\alpha}^{k\,2p}(G_0)}^{2} = \sum_{2pi_1+i_2\le 2pk} \iint_{G_0} t^{\alpha-(2kp-2i_1p-2i_2)/p} \left| \frac{\partial^{i_1+i_2}u}{\partial t^{i_1}\partial x^{i_2}} \right|^{2} \,dt\,dx . \]

An important role for us is played by the transformation

\[ t=e^{-2p\tau},\qquad x_i=\omega_i e^{-\tau}, \tag{8} \]

which maps the domain \(G_0\) into a cylinder \(D_0\), the equation of whose boundary \(D_0'\) has the form

\[ 1=\sum a_{i_1\ldots i_n}\omega_1^{i_1}\cdots \omega_n^{i_n}. \]

It is easy to verify that a function \(u\) from \(\dot H_{\alpha}^{k\,2p}(G_0)\), after the transformation (8), is transformed into a function \(u_1\) such that \(u_1e^{(-n-2p-2\alpha p+4kp)/2\tau}\) belongs to the Slobodetskii space \(H_2^{k\,2p}(D_0)\). For arbitrary \(k\) the norm of \(u\) in \(\dot H_{\alpha}^{k\,2p}(G_0)\) is defined to be equal to the norm of the function \(u_1e^{(-n-2p-2\alpha p+4kp)/2\tau}\) in \(H_2^{k\,2pk}(D)\). The space of boundary functions is introduced analogously: \(\dot H_{\alpha}^{k\,2p}(\Gamma_0)\). After the transformation (8) and the subsequent Fourier transform in \(\tau\), problem (6) passes into a certain problem:

\[ \widetilde L_0(\omega,\lambda,\partial/\partial\omega)\widetilde u = \widetilde F(\omega,\lambda), \]

\[ \widetilde B_{j0}(\omega,\lambda,\partial/\partial\omega)\widetilde u = \widetilde\Phi_j(\omega,\lambda) \tag{9} \]

in an infinitely smooth domain of the space \((\omega_1,\ldots,\omega_n)\). Using the results of [2], we obtain that there exists an operator \(R(\lambda)\) such that \(\widetilde u=R(\lambda)[\widetilde F,\widetilde\Phi_j]\), where \(R(\lambda)\) is a meromorphic function of \(\lambda\) and in each

in the strip \(|\operatorname{Im}\lambda|<\mathrm{const}\) it has a finite number of poles. With the help of \(R(\lambda)\) one can solve problem (1), (3) in the domain (2). This problem turns out to be solvable if on the line
\[ \operatorname{Im}\lambda=\frac{-n-2p-2\alpha p+4kp+4lp}{2} \]
\(R(\lambda)\) is regular. Further, using the Fourier transform, one solves problem (6) in the half-space \(x_n>0\) and in the whole space \((t,x)\).

After this we solve problem (1), (3) in a bounded domain by constructing a regularizer. For this purpose the surface (2) is reduced in a neighborhood of the point \((0,0)\) to the form (7) by the transformation
\[ t=t', \]
\[ x=x'_i+\sum_{s>1} t^{s/2p}\psi_{si}\left(\frac{x}{t^{1/2p}}\right). \]

This transformation preserves the form of equations (1), (3). In some cases (for example, when \(p=1\) or \(n=1\)) this transformation turns out to be infinitely differentiable. The regularizer can be constructed according to the schemes of work \((^2)\). For a solution of problem (1), (3) belonging to \(\dot H_\alpha^{k_1+2p}\), under the condition
\[ f\in \dot H_{\alpha_1}^{k,2p}(G),\qquad \varphi_j\in \dot H_{\alpha_1}^{\,k+l-m_j/2p-1/4p,\,2p}(\Gamma) \]
one can obtain, in a neighborhood of the point \((0,0)\), the representation
\[ u=\sum_{0<s<s_j} a_{jsq}t^{-i\lambda_j/2p}\ln^s t\, P_{sjq}\bigl(t\ln^q t\bigr)+w, \tag{10} \]
\[ \frac{-n-2p-2\alpha p+4k_1p+4lp}{2} <\operatorname{Im}\lambda_j< \frac{-n-2p-2\alpha_1p+4kp+4lp}{2}, \]
where \(\lambda_j\) are poles of \(R(\lambda)\) of multiplicity \(s_j\); \(P_{sjq}\) is a polynomial with coefficients—infinitely differentiable functions of \(x/t^{1/2p}\); \(a_{jsq}\) are constants depending on \(u\); and \(w\) is a function from the space \(\dot H_{\alpha_1}^{k+l,2p}(G)\). This representation can be used to study the smoothness of the solution.

The case of the Slobodetskii spaces \(H_2^{k,2pk}(G)\) is included in ours in the following way. If the function \(v(x,t)\) is such that
\[ \iint_G |v|^2\,dx\,dt+\iint_G t^\alpha \left|\frac{\partial^{k_1+k_2}v}{\partial t^{k_1}\partial x^{k_2}}\right|^2\,dt\,dx<\infty, \]
\[ 2pk_1+k_2=2pk;\quad \text{the number } -\,n/2-p-2\alpha p+4k \text{ is not an integer}, \]
then one can prove that the function \(v\) has the form
\[ v=p(x,t)+v_1, \]
where \(p(x,t)\) is a polynomial whose generalized order is \((2p-1)\), and \(v_1\in \dot H_\alpha^{k,2p}(G)\). After making the corresponding change of variables, one can pass to the spaces \(\dot H_\alpha^{k,2p}(G)\).

By investigating the location of the poles of the function \(R(\lambda)\), one can obtain, in concrete cases, various smoothness theorems.

In the case when problem (1), (3) is a parabolic semibounded problem (for the definition see work \((^2)\)), the poles of \(R(\lambda)\) are situated below some line \(\operatorname{Im}\lambda=K\). Using this, one can show that a sufficiently smooth solution of problem (1), (3) is infinitely differentiable. If the boundary conditions are Dirichlet conditions, and equation (1) has the form
\[ \frac{\partial u}{\partial t} =(-1)^p\sum_{i=0}^{2p} a_{i_1\ldots i_n}^{(i)} \frac{\partial^{2p-i}u}{\partial x_1^{i_1}\cdots \partial x_n^{i_n}} +f=Lu+f, \tag{11} \]
\[ \operatorname{Re}\sum a_{i_1\ldots i_n}^{(0)}(i\lambda_1)^{i_1}\cdots(i\lambda_n)^{i_n}>0,\qquad f\in H_2^{k,2pk}(G),\quad \text{then } u\in H_2^{k+1,\,2pk+2p}(G), \]

In this case it is shown that the function \(R(\lambda)\) has no poles in the upper complex half-plane. In the case of the analogous problem for the equation

\[ \frac{\partial u}{\partial t} = -(-1)^p \sum_{i=0}^{2p} a^{(i)}_{i_1\ldots i_n} \frac{\partial^{2p-i}}{\partial x_1^{i_1}\ldots \partial x_n^{i_n}} + f \]

the smoothness of the solution depends on the numbers \(a_{i_1\ldots i_n}\) entering into the equation of the boundary surface. It can be proved that the smoothness improves when the quantity

\[ \max\left( \sum_{i_1+\cdots+i_n=2pq} a_{i_1\ldots i_n}\lambda_1^{i_1}\cdots \lambda_p^{i_n} \Big/ \left(\lambda_1^{2pq}+\cdots+\lambda_n^{2pq}\right) \right) \]

is decreased.

In the case when the boundary conditions are Dirichlet conditions and \(f\) is a function from \(L_2(G)\), the problem was studied by V. P. Mikhailov in paper \({}^{1}\). The case when the domain contains no singular points was studied in paper \({}^{2}\).

Moscow State University
named after M. V. Lomonosov

Received
18 XII 1964

REFERENCES

\({}^{1}\) V. P. Mikhailov, Matem. sborn., 62, issue 2 (1963).
\({}^{2}\) M. S. Agranovich, M. I. Vishik, DAN, 149, No. 2 (1963).
\({}^{3}\) I. G. Petrovsky, Uch. zap. Moscow Univ., issue 2 (1934).