Abstract
Full Text
MATHEMATICS
A. Cifiligu, V. N. Maslennikova, and L. I. Kamynin
ON THE APPLICABILITY OF THE FOURIER METHOD TO THE SOLUTION OF THE FIRST BOUNDARY-VALUE PROBLEM FOR A QUASILINEAR PARABOLIC EQUATION
(Presented by Academician S. L. Sobolev on 10 IX 1959)
In this note we consider a parabolic equation with a nonlinearity of the form
[
\frac{\partial u}{\partial t}
=
\frac{\partial}{\partial x}
\left[
p(x)\frac{\partial u}{\partial x}
\right]
-
q(x)u
+
\mu f(x,t,u(x,t)).
\tag{1}
]
We assume that on the interval ([0,l]) the function (p(x)) is twice continuously differentiable, and (q(x)) is continuously differentiable, with (p(x)\ge p_0>0). It is assumed that (f(x,t,u)) is continuous together with its derivatives (\partial f/\partial x), (\partial f/\partial u), (\partial^2 f/\partial x^2), (\partial^2 f/\partial x\partial u), and (\partial^2 f/\partial u^2) with respect to all its arguments in the domain ((0\le x\le l,\ 0\le t\le T,\ |u|<\infty)); moreover, for (0\le x\le l), (0\le t\le T), and (|u|\le K), the inequalities
[
\left|\partial^k f/\partial x^m\partial u^n\right|
\le
F(T)\varphi(K),
\qquad
k=0,1,2,\quad m+n=k,
\tag{2}
]
hold, where (F(T)) and (\varphi(K)) are nonnegative, nondecreasing functions of their arguments, defined for all positive (T) and (K) and finite for finite (T) and (K). In addition, let (f(x,t,u)) satisfy
[
f(0,t,u)=f(l,t,u)=0.
\tag{3}
]
The first boundary-value problem for a non-self-adjoint parabolic equation with coefficients depending on (u) has been considered by a number of authors ((^{5-8})). The purpose of our note is to apply the Fourier method and Schauder’s method to the solution of the problem posed. As a result of applying the Fourier method, the problem is reduced to an infinite system of nonlinear integral equations, whose solution is obtained by Schauder’s method (the fixed-point method). Two definitions of generalized solutions are introduced; to prove their existence and uniqueness, fewer restrictions on the initial data are required than for classical solutions.
First consider the classical formulation of the first boundary-value problem for equation (1), consisting in the determination of a solution (u(x,t)), continuous together with the derivatives (\partial u/\partial x), (\partial u/\partial t), (\partial^2 u/\partial x^2) in the closed domain (G) ((0\le x\le l,\ 0\le t\le T)), satisfying the initial condition
[
u(x,0)=g(x)
\tag{4}
]
and the boundary conditions
[
u(0,t)=u(l,t)=0.
\tag{5}
]
By the Fourier method, the solution of the first boundary-value problem (1), (4), (5) is sought in the form
[
u(x,t)=\sum_{n=1}^{\infty} A_n(t)X_n(x),
\tag{6}
]
which reduces the problem (1), (4), (5) to the study of the infinite system of nonlinear integral equations
[
A_n(t)
=
c_n e^{-\lambda_n t}
+
\mu
\int_0^t\int_0^l
e^{-\lambda_n(t-\tau)}
f\left(x,\tau,\sum_{k=1}^{\infty} A_k(\tau)X_k(x)\right)
X_n(x)\,d\tau\,dx,
\tag{7}
]
(n=1,2,\ldots), where (X_n(x)) is a complete orthonormal system of eigenfunctions of the Sturm—Liouville problem
[
L(X)+\lambda X=0;
\tag{8}
]
[
X(0)=X(l)=0;
\tag{9}
]
[
L(X)\equiv \frac{d}{dx}\left[p(x)\frac{dX(x)}{dx}\right]-q(x)X(x);
\tag{10}
]
[
c_n=\int_0^l g(x)X_n(x)\,dx.
\tag{11}
]
In what follows we use the well-known ({}^{(1)}) properties of the eigenfunctions (X_n(x)) and eigenvalues (\lambda_n) of the Sturm—Liouville problem (8), (9).
Lemma 1. If (f(x,t,u)) satisfies (2), (3), and (u(x,t)) is measurable in (t) and continuous, together with (\partial u/\partial x), in (x), and in (G) satisfies the inequalities
[
|u(x,t)|\leq K;
\tag{12}
]
[
|\partial u(x,t)/\partial x|\leq K;
\tag{13}
]
[
|\partial u(x+\Delta x,t)/\partial x-\partial u(x,t)/\partial x|\leq K|\Delta x|,
\tag{14}
]
then the Fourier coefficients of (f(x,t,u(x,t))) with respect to the system (X_n(x)) satisfy the inequality
[
|b_n(t)|\leq D/\lambda_n,
\tag{15}
]
where the constant (D) depends on (p(x)), (p'(x)), (q(x)), (K), (T).
The investigation of system (7) is carried out according to the scheme of the work ({}^{(2)}) of one of the authors as follows. Consider the nonmetrizable complete linear topological space (C^{(1)}) of functions (u(x,t)) satisfying (5), continuous in (x) and (t) in the domain (G), having partial derivatives (\partial u/\partial x), continuous in (x) and measurable in (t), with topology given by the operation of passage to the limit
[
\lim_{k\to\infty}\sup_{(x,t)\in G}|u(x,t)-u_k(x,t)|=0,
\tag{16}
]
[
\lim_{k\to\infty}\sup_{0\leq x\leq l}|\partial u(x,t)/\partial x-\partial u_k(x,t)/\partial x|=0
\tag{17}
]
(for any fixed (t) from ([0,T])). In (C^{(1)}) consider the closed convex set (C_k^{(1)}) of elements of (C^{(1)}) satisfying (12), (13), and (14), where (K) is a constant. On (C_k^{(1)}) define a nonlinear operator (A) ((v=Au)) in such a way that, if (u\in C_k^{(1)}) and is given with the aid of (6), then
[
v(x,t)=\sum_{n=1}^{\infty} B_n(t)X_n(x),
]
where (B_n(t)) is determined with the aid of (u(x,t)) from
[
B_n(t)=c_ne^{-\lambda_n t}+\mu\int_0^t\int_0^l e^{-\lambda_n(t-\tau)}f(x,\tau,u(x,\tau))X_n(x)\,d\tau\,dx,\qquad n=1,2,\ldots
]
The range of values of the operator (A) then lies in (C^{(1)}).
Theorem 1. For sufficiently small (\mu), the operator (A) maps the convex set (C_k^{(1)}) in (C^{(1)}) into its compact part.
Theorem 2. If (f(x,t,u)) satisfies (2), (3), then, for (\mu) sufficiently small, there exists at least one invariant point of the mapping (v=Au).
Theorem 3. The infinite system (7) has a unique solution (A_n(t)), (n=1,2,\ldots), in the class of continuous functions satisfying the conditions (|A_n(t)|\leq Aa_n), where (\sum_{n=1}^{\infty} a_n<\infty), if (f(x,t,u)) is continuous in (x), (t), and satisfies a Lipschitz condition in (u).
Theorem 4. If (p(x), q(x), f(x,t,u)) satisfy the requirements formulated at the beginning of the note; (g(x)) is three times continuously differentiable and satisfies conditions (9) and (L(g)=0) for (x=0) and (x=l), then, for sufficiently small (\mu), there exists in (G) a unique classical solution of the first boundary-value problem (1), (4), (5), representable in the form (6).
Remark. Since we impose no restrictions on the character of the behavior of (\partial f/\partial u), it is not surprising (see (3)) that the existence and uniqueness of the classical solution of the first boundary-value problem (1), (4), (5) have been proved for small (\mu).
Theorem 5 (correctness). Classical solutions (u_1(x,t), u_2(x,t)) of the first boundary-value problem for equation (1), satisfying (5), (12), and the initial conditions
[
u_1(x,0)=g_1(x);
\tag{18}
]
[
u_2(x,0)=g_2(x)
\tag{19}
]
for sufficiently small (\mu) will differ in modulus by an arbitrarily small amount in (G), if (|g_1(x)-g_2(x)|,\ |g'_1(x)-g'_2(x)|) are sufficiently small for all (x) on ([0,l]).
Theorem 6. If (u_1(x,t), u_2(x,t)) are classical solutions in (G) of the first boundary-value problem (1), (5), representable in the form (6), satisfying (12) and the initial conditions (19), (20), then for (0\leq t\leq T)
[
\int_0^l [u_1(x,t)-u_2(x,t)]^2\,dx
\leq
A\int_0^l [g_1(x)-g_2(x)]^2\,dx,
\tag{20}
]
where the constant (A) depends only on (K) and (T).
Definition 1. A continuous function (u(x,t)) will be called a generalized solution of the first boundary-value problem (1), (4), (5), if (u(x,t)) is the limit, uniformly convergent in (G) as (m\to\infty), of a sequence of classical solutions (u_m(x,t)), representable in the form (6), of equation (1) with boundary conditions (5) and initial data
[
u_m(x,0)=g_m(x)
\tag{21}
]
such that (g_m(x)) converges uniformly on ([0,l]) to (g(x)).
Theorem 7. If (g(x)) satisfies (9) and has (g'(x), g''(x)), with (g''(x)) square-integrable on ([0,l]), then, for sufficiently small (\mu), there exists at least one generalized solution (u(x,t)), in the sense of Definition 1, of the first boundary-value problem (1), (4), (5).
To prove Theorem 7, one constructs a sequence of classical solutions (u_m(x,t)) of problem (1), (5), satisfying the initial conditions (21), where
[
g_m(x)=\sum_{k=1}^{m} c_k X_k(x)
]
((c_k) are the Fourier coefficients of (g(x)) (11)). The classical solutions are sought in the form
[
u_m(x,t)=\sum_{k=1}^{\infty} A_k^{(m)}(t)X_k(x),
]
which leads to the question of the existence of a solution of a finite system of nonlinear integral equations. A linear nonmetrizable complete topological space (R) of functions (u(x,t)), defined on (G), continuous in (x), measurable in (t), satisfying (5), is introduced, with topology specified by the limiting transition
[
\lim_{k\to\infty}\sup_{0\leq x\leq l}|u(x,t)-u_k(x,t)|=0.
]
In (R) one considers a compact convex set (D) of elements (u) from (R), satisfying (12) and (|u(x+\Delta x,t)-u(x,t)|\leq L|\Delta x|). On (D)
one considers the operator (A_m) ((v=A_m u)), defined as follows: if (u\in D), then put
[
v(x,t)=\sum_{k=1}^{m} B_k^{(m)}(t)X_k(x),
]
where (B_k^{(m)}(t)) is determined with the help of (u(x,t)) from
[
B_k^{(m)}(t)=c_k e^{-\lambda_k t}
+\mu\int_0^t\int_0^l e^{-\lambda_k(t-\tau)} f(x,\tau,u(x,\tau))X_k(x)\,d\tau\,dx,
\quad k=1,2,\ldots,m.
]
The proof of the existence of a solution of the equation (u=A_m u) in the complete linear topological, but nonmetrizable, space (R) reduces to the application of Schauder’s principle in a complete linear metric space by means of a device of A. N. Tikhonov ((^4)).
Theorem 8 (correctness of the generalized solution). If there exists a generalized solution, in the sense of Definition 1, of the first boundary-value problem (1), (4), (5), then it is unique in the class of functions continuous on (G), provided the initial function is continuous on ([0,l]) and satisfies (9).
Theorem 9. If (u_1(x,t)), (u_2(x,t)) are generalized, in the sense of Definition 1, solutions of the first boundary-value problem (1), (5) with initial data (18), (19), then, for (\mu) sufficiently small, (|u_1(x,t)-u_2(x,t)|) will be arbitrarily small in (G), provided (|g_1(x)-g_2(x)|), (|g_1'(x)-g_2'(x)|) are sufficiently small for all (x) in ([0,l]).
Definition 2. A bounded function (u(x,t)), for which
[
\int_0^l u^2(x,t)\,dx
]
is continuous with respect to (t) on ([0,T]), will be called a generalized solution in (G) of the first boundary-value problem (1), (4), (5), if in (G) there exists a uniformly bounded sequence of classical solutions (u_m(x,t)), representable in the form (6), of the first boundary-value problem (1), (5) with initial data (21) such that
[
\lim_{m\to\infty}\int_0^l [g(x)-g_m(x)]^2\,dx=0
]
and, uniformly in (t) ((0\le t\le T)),
[
\lim_{m\to\infty}\sup_{0\le t\le T}\int_0^l [u(x,t)-u_m(x,t)]^2\,dx=0.
]
Theorem 10. If there exists a generalized, in the sense of Definition 2, solution (u(x,t)) in (G) of the first boundary-value problem (1), (4), (5), then it is unique in the class of bounded functions (v(x,t)) for which
[
\int_0^l v^2(x,t)\,dx
]
is continuous with respect to (t). The initial function (g(x)) is here assumed continuous on ([0,l]) and satisfying (9).
Theorem 11. If (g(x)) satisfies (9) and is continuously differentiable, then, for (\mu) sufficiently small, there exists at least one generalized, in the sense of Definition 2, solution (u(x,t)) of the first boundary-value problem (1), (4), (5), and moreover (|u(x,t)|\le K). For this generalized solution the inequality (20) also holds.
Steklov Mathematical Institute
Academy of Sciences of the USSR
Tirana State University
Tirana, Albania
Received
10 IX 1959
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