Abstract
Full Text
UDC 517.946
MATHEMATICS
T. G. Pletneva, S. D. Eidelman
THEOREMS ON THREE CYLINDERS FOR SOLUTIONS OF LINEAR EVOLUTION QUASI-ELLIPTIC EQUATIONS
(Presented by Academician S. L. Sobolev on 11 XI 1969)
Here, with the aid of special algebraic conditions, we single out classes of evolution equations of arbitrary order containing, for example, besides parabolic equations, equations of the type of equations of transverse vibrations of elastic plates and rods, for which (L_1)-estimates of solutions are obtained that do not contain the values of the solutions on the initial hyperplane, and, in the case of cylinders adjacent to the boundary, the values of the solutions on the boundary hypersurface. The investigation covers equations having power singularities on the initial hypersurface (boundary hypersurface).
The methods used by us are very close to those set forth in ((^1)) and are their natural continuation and development.
1. Conditions. Notation. Consider the equation
[
\mathscr{L}u \equiv \mathscr{L}\left(t,x;\frac{\partial}{\partial t},D_x\right)u
\equiv
\sum_{k_0p+|k|\le m} a'_{k_0k}(t,x)
\frac{\partial^{k_0}}{\partial t^{k_0}}D_x^k u(t,x)
= f(t,x).
\tag{1}
]
We introduce a number of conditions necessary for the formulation of the results:
A. The operator (\mathscr{L}) has the Lagrange adjoint operator
[
\mathscr{L}^*\left(t,x;\frac{\partial}{\partial t},D_x\right)
\equiv
\sum_{k_0p+|k|=m} a_{k_0k}(t,x)
\frac{\partial^{k_0}}{\partial t^{k_0}}D_x^k
+
\sum_{k_0p+|k|<m} a_{k_0k}(t,x)
\frac{\partial^{k_0}}{\partial t^{k_0}}D_x^k
\equiv
]
[
\equiv P_0(t,x;\partial/\partial t,D_x)
+
P_1(t,x;\partial/\partial t,D_x);
\qquad
p \ge 1;\quad \nu=m/p.
]
B. The polynomials
[
\text{B}_1.\quad P_0(t,x;0,\sigma),
]
[
\text{B}2.\quad P_0(t,x;\sigma^p,\sigma).
]
[
\text{B}{3,\beta}.\quad
Q_0(t,x,\sigma')
=
\sum}^{\nu
\prod_{l=1}^{\mu-1}\left(1-\frac{l}{\beta}\right)
\frac{1}{\mu!}
\left.
\frac{\partial^\mu}{\partial\lambda^\mu}
P_0(t,x;\lambda,\sigma)
\right|{\lambda=0}
\sigma,}^{p\mu
]
[
p\ge \nu,\qquad
\sigma'=(\sigma_{n+1},\sigma_1,\ldots,\sigma_n).
]
[
\text{B}4^{(\alpha)}.\quad
T_0^{(\alpha)}(t,x,\sigma')
\equiv
\sum}^{\nu
\prod_{l=1}^{m-p\mu-1}\left(1-\frac{l}{\alpha}\right)
\frac{1}{\mu!}
\left.
\frac{\partial^\mu P_0}{\partial\lambda^\mu}
\right|{\lambda=0}
\sigma,}^{p\mu
\qquad \alpha\ge m.
]
[
\text{B}5^{(\gamma)}.\quad
Q_0^{(\gamma)}(t,x;\sigma')
\equiv
\sum}^{m
\prod_{l=1}^{q-1}\left(1-\frac{l}{\gamma}\right)
\frac{1}{q!}
\left.
\frac{\partial^q P_0}{\partial\sigma_n^q}
\right|_{\sigma_n=0}
\sigma_n^q,
\qquad \gamma\ge m.
]
[
\text{B}{6,\beta}^{(\gamma)}.\quad
R(t,x;\sigma')}^{(\gamma)
\doteq
\sum_{\mu=0}^{\nu}\sum_{q=0}^{m}
\prod_{l=1}^{\mu-1}\left(1-\frac{l}{\beta}\right)
\prod_{s=1}^{q-1}\left(1-\frac{s}{\gamma}\right)
\frac{1}{\mu!q!}\times
]
[
\times
\left.
\frac{\partial^{\mu+q}P_0}
{\partial\lambda^\mu\partial\sigma_n^q}
\right|{\sigma_n=0,\lambda=0}
\sigma\sigma_n^q}^{p\mu
]
are uniformly elliptic with constant uniform ellipticity (\delta_0,\delta_1,\delta_\beta,\delta^{(\alpha)},\delta_1^{(\gamma)},\delta_\beta^{(\gamma)}), respectively.
C. The ranges of values of the polynomials
[
\text{C}1.\ P_0(t,x;0,\sigma).\qquad
\text{C}_2.\ P_0(t,x;\sigma^p,\sigma).\qquad
\text{C}{3,\beta}.\ Q(t,x;\sigma').
]
[
\text{C}4^{(\alpha)}.\ T_0^{(\alpha)}(t,x;\sigma').\qquad
\text{C}_5^{(\gamma)}.\ Q_0^{(\gamma)}(s,x;\sigma').\qquad
\text{C}(t,x;\sigma')}^{(\gamma)}.\ R_{0\beta}^{(\gamma)
]
for any real (\sigma') lies in the cone (sector) (K_{\varphi_1}:\ |\arg z|\leq \varphi_1<\pi/2) of the complex (z)-plane.
E. The coefficients (a_{k_0k}(t,x)) of the polynomial (\mathscr L^*(t,x;\lambda,\sigma)):
E(1). Are uniformly bounded by a constant (e_1).
E(_2). For (k_0p+|k|=m) they are uniformly continuous.
E(_3). (a), (b\in(0,1)), are uniformly bounded by a constant (e_3).}(bt,b^{1/p}x)b^{(m-k_0p-|k|)/p
Condition B(1) means ellipticity of the spatial part of the operator (\mathscr L); condition B(_2) will be called the quasi-ellipticity of this operator. Let us note that, for sufficiently large (\beta;\alpha;\beta) and (\gamma), conditions B(), C({3,\beta}), B(_4^{(\alpha)}), C(_4^{(\alpha)}), B() follow from conditions B(_2), C(_2), respectively. If the coefficients of the polynomial (P_0(t,x;\lambda,\sigma)) are real, then conditions C (with (\varphi_1=0)) follow from conditions B. It is useful to note that for (\beta=\nu,\ \gamma=m):}^{(\gamma)}), C(_{5,\beta}^{(\gamma)
[
Q_{0\nu}=\sum_{\mu=0}^{\nu} C_\nu^\mu
\frac{\partial^\mu}{\partial\lambda^\mu}P_0\bigg|{\lambda=0}\sigma,}^{p\mu
\qquad
Q_0^{(m)}=\sum_{q=0}^{m} C_m^q
\frac{\partial^q}{\partial\sigma_n^q}P_0\bigg|_{\sigma_n=0}\sigma_n^q;
]
[
R_{0\nu}^{(m)}=\sum_{\mu=0}^{\nu}\sum_{q=0}^{m} C_\nu^\mu C_m^q
\frac{\partial^{\mu+q}}{\partial\lambda^\mu\partial\sigma_n^q}P_0
\bigg|{\lambda=0,\sigma_n=0}\sigma\sigma_n^q .}^{p\mu
]
In the case of an equation of first order in (t):
[
P_0(t,x;\lambda,\sigma)=\lambda+\sum_{|k|=m} a_k(t,x)\sigma^k,\qquad
\nu=1,\quad m=p,
]
[
Q_{01}(t,x;\sigma')\equiv P_0(t,x;\sigma_{n+1}^{p},\sigma)
=\sigma_{n+1}^m+\sum_{|k|=m} a_k(t,x)\sigma^k,
]
[
R_{01}^{(m)}(t,x;\sigma')=Q_0^{(m)}(t,x;\sigma')
=\sigma_{n+1}^{m}+\sum_{q=0}^{m} C_m^q
\frac{\partial^q}{\partial\sigma_n^q}P_0(t,x;0,\sigma)\bigg|_{\sigma_n=0}\sigma_n^q .
]
In particular, for (m=2),
[
Q_{01}=\sigma_{n+1}^2+\sum_{i,j=1}^{n} a_{ij}(t,x)\sigma_i\sigma_j,
]
(a_{ij}(t,x)) are real functions, and conditions B(5^{(2)}), B()_1^n).}^{(2)}) mean the ellipticity of the polynomials (Q_0^{(2)}), (R_{01}^{(2)}); the best condition invariant under rotations in the space ((x_1,x_2,\ldots,x_n)) for this is the inequality (\lambda_1<(n+2+2\sqrt2)\lambda_n), where (\lambda_1) is the largest and (\lambda_n) the smallest characteristic number of the matrix ((a_{ij
If the polynomial (P_0(t,x,\sigma_{n+1}^{p},\sigma)) with real positive coefficients contains only even powers of (\sigma_{n+1},\sigma_1,\sigma_2,\ldots,\sigma_n), then all the conditions described above are automatically satisfied. In particular, they are satisfied for the polynomial
[
\sigma_{n+1}^2+a^2(\sigma_1^2+\cdots+\sigma_n^2)^2,
]
corresponding to the equation of transverse elastic vibrations. If the operator has the form
[
\mathscr Lu=\sum_{k_0p+|k|\leq m}(-1)^{k_0+|k|}
\frac{\partial^{k_0}}{\partial t^{k_0}}D_x^k\bigl(a_{k_0k}(t,x)u\bigr),
]
then it is sufficient to require only boundedness of the coefficients (a_{k_0k}(t,x)).
Define the cone (sector) (K) of the complex (z)-plane by the inequality
[
K=K_{\varphi_2}={z;\ |\arg z|\leq \varphi_2<\pi/2-\varphi_1}.
]
With the aid of the cone (K) we represent the function (u(t,x)) in the form (u(t,x)=u^+(t,x)-u^-(t,x)), where (u^+(t,x)=u(t,x)) if (u\in K); (u^+(t,x)=0) if (u\notin K). Denote
[
\Pi_{(\tau,T),R}={(t,x),\ \tau<t<T,\ |x|\eta},\qquad
\Pi_{(0,T),R}^{(0)}=\Pi_{T,R}^{+};
]
[
\Pi_{(0,T),R}=\Pi_{T,R};\qquad
M(r)=r\ \text{for } r\in[0,1);\qquad
M(r)=1\ \text{for } r\geq1,
]
[
|u|{(\tau,T),R;\beta}=
\iint M[t^\beta]|u|\,dt\,dx;}
]
[
{u}{(\tau,T),R,\eta;\beta,\gamma;q}=
\iint)^q]\,M[t^\beta]\,M[x_n^\gamma]|u|\,dt\,dx;}^{(\eta)}} M[(t^\nu+|x|^{m
]
[
|u|{(0,T),R;\beta}=|u|;
\qquad
{u}{(0,T),R,0;\beta,\gamma,q}={u};
]
[
\langle u\rangle_{(-T_1,T),R;\alpha}
=
\iint_{\Pi_{(-T_1,T),R}} (R^2-x^2)^\alpha |u|\,dt\,dx.
]
2. Theorems on three cylinders
Theorem 1. The conditions (A_1, B_1, C_1) are satisfied in (\overline{\Pi}{T,2}), (B}, C_{3,\beta}) in (\overline{\Pi{\varepsilon_0,2}); (E_3) in (\overline{\Pi}},\,2b^{-1/p}}) (if one assumes (E_2) in (\overline{\Pi{T,2}), then (C_1) is superfluous, and (B) need only be assumed for (t=0)).}, C_{3,\beta
Let (u(t,x)) be a weak solution of equation (1) in (\Pi_{T,2}). Then there exist positive constants (h_1,b_1,\lambda_2), depending only on (n,m,p,\delta_0,\delta_\beta,\varphi_1,e_3,\varepsilon_0), such that for every (h\in(0,h_1)) the estimate holds
[
|u|{T-h,\,1+b_1h^{1/p};\,\beta-\nu}
\le
\lambda_1\bigl(|u|
+|u^{-}|{T,2;\,\beta-\nu}
+|f|\bigr).
]
Theorem 2. The conditions (A,E_1) are satisfied in (\Pi_{(-T_1,T),R}); (B_4^{(\alpha)}, C_4^{(\alpha)}) in (\Pi_{(-T,0],R}) (if one assumes (E_2) in (\Pi_{(-T_2,0],R}), then (B_4^{(\alpha)}, C_4^{(\alpha)}) need only be assumed for (t=0)).
Let (u(t,x)) be a weak solution of equation (1) in (\Pi_{(-T_1,T),R}); then there exist positive constants (t_0,\lambda_2), depending only on (n,m,p,\delta^{(\alpha)},\varphi_1,e_1,\varepsilon_0,R), such that
[
\langle u\rangle_{(-t_0,T),R;\alpha-m}
\le
\lambda_2\bigl(
\langle u\rangle_{T,R;\alpha-m}
+
\langle u^{-}\rangle_{(-T_1,T),R;\alpha-m}
+
\langle f\rangle_{(-T_1,T),R,\alpha}
\bigr).
]
Theorem 3. The conditions (A,B_1,C_1) are satisfied in (\overline{\Pi}^{+}{T,2}); (B},C_{3,\beta}) in (\overline{\Pi}^{+{\varepsilon_0,2}), (B_5^{(\gamma)},C_5^{(\gamma)}) in (\overline{\Pi}^{+}}\cap{x_n\in[0,\varepsilon_0]}); (B_{6,\beta}^{(\gamma)},C_{6,\beta}^{(\gamma)}) in (\overline{\Pi}^{+{\varepsilon_0,2}\cap{x_n\in[0,\varepsilon_0]}); (E_3) in (\overline{\Pi}^{+}},\,2b^{-1/p}}) (if (E_2) is assumed in (\overline{\Pi}^{+{T,2}), then (C_1) is superfluous, (B) for (t=0,\ x_n=0)).},C_{3,\beta}) need only be assumed for (t=0); (B_5^{(\gamma)},C_5^{(\gamma)}) for (x_n=0); (B_{6,\beta}^{(\gamma)},C_{6,\beta}^{(\gamma)
Let (u(t,x)) be a weak solution of equation (1) in (\Pi^{+}{T,2}); then there exist positive constants (h_2,b_2,\lambda_3), depending only on (n,m,p,\delta_0,\delta_1^{(\gamma)},\delta\beta^{(\gamma)},\delta_\beta,\varphi_1,\varepsilon_0,e_3), such that for every (h\in(0,h_2)) the estimate holds
[
{u}{T-h,\,1+b_2h^{1/p};\,\beta-\nu,\,\gamma-m;\,1}
\le
\lambda_3\bigl(
{u}
+
{u^{-}}{T,2;\,\beta-\nu,\,\gamma-m;\,1}
+
{f}
\bigr).
]
The proofs of Theorems 1 and 3 are carried out by covering the initial cylinder with lunettes, semi-lunettes, and quarters of lunettes, in which the norm of the positive part is estimated, under the assumptions made, on the basis of the “sign-definiteness” of (\mathcal L^*\Phi); as (\Phi) one takes, respectively, functions of the form
[
t^\beta(1-x^2-t^2)^\alpha,\qquad
t^\beta x_n^\gamma(1-x^2-t^2)^\alpha,\qquad
x_n^\gamma(1-x^2-t^2)^\alpha,
]
while Theorem 2 is proved by covering the whole cylinder (\Pi_{T,R}) from below by a cylinder of small height and using the function
[
t^\beta(R^2-x^2)^\alpha.
]
We note that for (\beta=\nu,\ \alpha=m,\ \gamma=m), Theorems 1 and 2 give (L_1)-estimates of solutions; for (\beta>\nu,\ \alpha>m,\ \gamma>m), they give estimates in a space with the corresponding weight. In this case the solutions are allowed to have singularities on the initial and boundary hyperplanes of arbitrary power order (for arbitrary (\beta,\alpha,\gamma)). If (\beta,\alpha,\gamma) are large, then all the estimates are valid under the assumption of quasellipticity of the polynomial (P_0(t,x;\lambda,\sigma)) and the fulfillment of condition (C_2).
Theorems 1, 2, and 3 have various applications. We shall present some of them.
3. Membership of positive solutions of quasi-elliptic equations in the space (L_1^{(\beta,\gamma)}(Q_T))
Let (Q_T=(0,T)\times\Omega), where (\Omega) is a bounded domain with boundary (S) belonging to the class (C^{m-l+n}) ((2)). Denote by (\nu(x)) the inward normal to (S) at the point (x). Then, as usual, the polynomial (P_0(t,x;\sigma_{n+1}^p,\sigma)) can be written in coordinates corresponding to the half-space (\nu\cdot x>0) in the form
[
P_0(t,x;\sigma_{n+1}^p,\theta'+\nu\theta_n)
\equiv
P_{0\nu}(t,x;\sigma_{n+1}^p,\theta),
\qquad
\theta'=(\theta_1,\ldots,\theta_{n-1},0),
]
and from the polynomial (P_{0\nu}) define (Q_{0\nu}^{(\gamma)}, R_{0\nu,\beta}^{(\gamma)}); the conditions (B) and (C) for these polynomials are assumed to hold uniformly in (\nu). Let (\Omega_{\varepsilon_0}) denote the set of points of the domain (\Omega) lying at distance greater than (\varepsilon_0) from (S), (Q_{(\varepsilon_0,T),\varepsilon_0}=)
(=(\varepsilon_0,T)\times \Omega_{\varepsilon_0}), (\rho(x)) is the distance from the point (x) to (S). Introduce the space (L_1^{(\beta,\gamma,q)}(Q_T)) of functions for which the norm
[
{u}{Q_T;\beta,\gamma}
=
\iint
M\bigl[(t^\nu+\rho(x)^m)^q\bigr]\,
M[t^\beta]\,
M[\rho(x)^\gamma]\,
|u(t,x)|\,dt\,dx
]
is finite.
With the aid of Theorems 1 and 3 one establishes
Theorem 4. Suppose that conditions (A), (B_1), (C_1), (E_1) are satisfied in (\overline Q_T); conditions (B_{3,\beta}), (C_{3,\beta}) in (\overline Q_{\varepsilon_0}); (B_5^{(\gamma)}), (C_5^{(\gamma)}) in (\overline Q_T/Q_{(\varepsilon_0,T),\varepsilon}); (B_5^{(\gamma)}), (C_5^{(\gamma)}) in (\overline Q_{\varepsilon_0}\cap{x_n\in[0,\varepsilon_0)}). If:
1) (u(t,x)) is a weak solution of equation (1) in (Q_T); 2) (f(t,x)\in L_1^{(\beta,\gamma;0)}(Q_T));
3) (u^{-}(t,x)\in L_1^{(\beta-\nu,\gamma-m;1)}(Q_T)), then (u(t,x)\in L_1^{(\beta-\nu,\gamma-m;1)}(Q_{T_1})) for any (T_1<T), and the estimate
[
{u}{Q};\beta-\nu,\gamma-m;1
\le
\lambda_4\bigl(
{u}{Q}
+
{u^{-}}{Q_T;\beta-\nu,\gamma-m;1}
+
{f}
\bigr)
]
is valid; (\lambda_4) depends only on (\lambda) (from Theorem 3), (\varepsilon_0), and (T-T_1).
With the aid of Theorems 1 and 3 one establishes propositions on the growth of solutions defined in an unbounded cylinder (where the coefficients may grow with the growth of the spatial coordinates), generalizing and strengthening the results stated in (1). This makes it possible, in particular, on the basis of the usual uniqueness theorems, to establish theorems on the coincidence of solutions with identical initial conditions under estimates of their negative components.
4. On solutions in an infinite (in (t)) tube
On the basis of Theorem 2 one establishes
Theorem 5. Suppose that conditions (A), (E_1), (B_4^{(\alpha)}), (C_4^{(\alpha)}) are satisfied in (\Pi_{(-\infty,0),R}). If (u(t,x)) is a weak solution of the equation (\mathscr L u=0) in (\Pi_{(-\infty,0),R}), (\langle u\rangle_{(-1,0),R;\alpha-m}<M_1), (\langle u^{-}\rangle_{(t,0),R;\alpha-m}\le M_2 e^{c|t|}), (t<0), then
[
\langle u\rangle_{(t,0),R;\alpha-m}
\le
K_1(c,\lambda_2)(M_1+M_2)e^{(C+\ln 2\lambda_2/t_0)|t|},
]
where (\lambda_2,t_0) are from Theorem 2.
If the coefficients (P_0(t,x;\sigma_{n+1}^{p},\sigma)) are real, then for sufficiently large (\alpha) all the conditions of Theorem 5 reduce to its ellipticity. If (P_0(0,\sigma)) is an elliptic polynomial, then the ellipticity of (P_0(\sigma_{n+1}^{p},\sigma)) is not only sufficient but also necessary in order that a positive solution of equation (1) with the principal group of terms could grow no faster than (e^{c|t|}) as (t\to-\infty).
5. Degenerate equations
All the theorems remain valid for degenerate equations. The auxiliary functions and the method of proof are the same. The algebraic conditions are derived in an analogous way. For illustration, let us write down the construction of the equation in the case of degeneration on a hyperplane. The operator (\mathscr L^*) is assumed to be of the form:
[
\mathscr L^{*}
=
\sum_{j=0}^{m} t^{-j/p}
\sum_{k_0p+|k|=m-j}
a_{k_0k}(t,x)\frac{\partial^{k_0}}{\partial t^{k_0}}D_x^k
+
P_1
\equiv
\sum_{j=0}^{m} t^{-j/p}P_{0j}+P_1,
]
the (a_{k_0k}(t,x)) are bounded; the coefficients of the polynomial (P_1) have the property that
(b_{k_0k}(t,x)t^{(m-|k_0k|)/p-k_0-\varepsilon_1}), (\varepsilon_1>0), are bounded. The principal polynomial is
[
Q_{0\beta}
=
\sum_{\mu=0}^{m}
\beta^{-j/p}
\prod_{s=1}^{\mu-1}
\left(1-\frac{s}{\beta}\right)
\frac{1}{\mu!}
\left.
\frac{\partial^\mu}{\partial\lambda^\mu}P_{0j}
\right|{\lambda=0}
\sigma.}^{p\mu+j
]
Kiev Polytechnic Institute
named after the 50th Anniversary of the Great October
Socialist Revolution
Received
23 X 1969
CITED LITERATURE
- V. A. Kondrat’ev, S. D. Eidel’man, DAN, 184, No. 5 (1969); 189, No. 3 (1969).
- V. A. Solonnikov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 83 (1965).