Mixed and boundary value problems for parabolic equations and systems

Problems of finding solutions

$$ u ( x , t ) = ( u _ {1} ( x , t ) \dots u _ {m} ( x ,\ t ) ) $$

in a domain $ D $ of a Euclidean space $ \mathbf R ^ {n+} 1 $( with points $ ( x , t ) = ( x _ {1} \dots x _ {n} , t ) $) of a parabolic system of equations or, when $ m = 1 $, of a parabolic equation satisfying additional conditions on some part of the boundary $ \partial D $ of the domain $ D $.

Let $ \Omega $ be a domain in $ \mathbf R ^ {n} $ with sufficiently smooth boundary $ \partial \Omega $ and let $ D $ be a cylinder $ \{ {x \in \Omega } : {0 < t < T } \} $ with lateral surface $ \Gamma = \{ {x \in \partial \Omega } : {0 < t < T } \} $, lower base $ \Omega _ {0} = \{ {x \in \Omega } : {t = 0 } \} $ and upper base $ \Omega _ {T} = \{ {x \in \Omega } : {t = T } \} $. The mixed Petrovskii problem for a linear parabolic system

$$ \tag{1 } u _ {t} + \sum _ {| \alpha | \leq 2 p } A _ \alpha ( x , t ) D _ {x} ^ \alpha u = f ( x , t ) ,\ ( x , t ) \in D , $$

$$ f ( x , t ) = ( f _ {1} ( x , t ) \dots f _ {m} ( x , t ) ) , $$

in the cylinder $ D $ consists of finding solutions of this system satisfying the initial conditions

$$ \tag{2 } u \mid _ {\Omega _ {0} } = \phi ( x) , $$

where $ \phi ( x) = ( \phi _ {1} ( x) \dots \phi _ {m} ( x) ) $, and the boundary condition

$$ \tag{3 } \left . B \left ( x ,t , \frac \partial {\partial x } \right ) u \right | _ \Gamma = \psi ( x , t ) , $$

where $ \psi ( x , t ) = ( \psi _ {1} ( x , t ) \dots \psi _ {p} ( x , t ) ) $ and $ B ( x , t , \partial / \partial x ) $ is a rectangular matrix with

$$ B _ {ij} \left ( x , t , \frac \partial {\partial x } \right ) = \ \sum _ {| \alpha | \leq q _ {i,j} } b _ \alpha ^ {i,j} ( x , t ) D _ {x} ^ \alpha , $$

$$ i = 1 \dots m ; \ j = 1 \dots p . $$

Suppose that the system is uniformly parabolic.

A classical solution of the mixed problem (1)–(3) is a vector function $ U ( x , t ) $ belonging to

$$ C _ {x,t} ^ {2p,1} ( D) \cap C _ {x,1} ^ {q,0} ( D \cup \Gamma ) \cap C ( D \cup \Gamma \cup \Omega bar _ {0} ) , $$

where $ q = \max q _ {i,j} $ for $ 1 \leq i \leq m $, $ 1 \leq j \leq p $, and satisfying (1) in $ D $ and conditions (2) and (3) on $ \Omega _ {0} $ and $ \Gamma $, respectively. Sometimes one considers more general solutions than this. In particular, one may drop the requirement that the solution be continuous at the points of $ \overline \Gamma \; \cap \overline \Omega \; _ {0} $ and replace it by the condition that it is bounded in $ D $.

If the complementarity (or Lopatinskii) condition holds (and if, for the sake of simplicity, $ \Omega $ is assumed to be bounded), then for sufficiently smooth data (the coefficients in (1) and (3) and the vector functions $ f $, $ \phi $ and $ \psi $) and under certain compatibility conditions, a classical solutions exists and is unique.

The basic mixed problems for a general linear uniformly-parabolic second-order equation

$$ \tag{1'} u _ {t} - L u \equiv u _ {t} - \sum _ {i , j = 1 } ^ { n } ( a _ {ij} ( x , t ) u _ {x _ {i} } ) _ {x _ {j} } + $$

$$ - \sum _ {i = 1 } ^ { n } b _ {i} ( x , t ) u _ { x _ {i} } - c ( x , t ) u = f ( x , t ) ,\ ( x , t ) \in D , $$

$$ a _ {ij} ( x , t ) = a _ {ji} ( x , t ) ,\ i , j = 1 \dots n , $$

are those of finding solutions of (1'}) that satisfy the initial condition

$$ \tag{2'} u \mid _ {\Omega _ {0} } = \phi ( x) $$

and one of the boundary conditions

$$ \tag{4 } u \mid _ \Gamma = \psi ( x , t ) , $$

the first mixed problem,

$$ \tag{5 } \left . \frac{\partial u }{\partial \nu } \right | _ \Gamma = \psi ( x , t ) , $$

the second mixed problem, or

$$ \tag{6 } \left . \left ( \frac{\partial u }{\partial N } + \sigma ( x , t ) u \right ) \right | _ \Gamma = \psi ( x , t ) , $$

the third mixed problem, where $ N $ is a co-normal of the elliptic operator $ L $.

Each of these problems satisfies the complementarity condition and, consequently, when the data are sufficiently smooth and the compatibility conditions hold, each has a classical solution. This solution can be obtained by the method of potentials, the method of finite differences, the Galerkin method, or, in the case when the functions $ a _ {i,j} $( $ i , j = 1 \dots n $), $ c $ and $ \sigma $ do not depend on $ t $ and $ b _ {i} \equiv 0 $, $ i = 1 \dots n $, by the Fourier method. For example, in order to solve the first mixed problem for equation (1) it is sufficient to require that the coefficients of the equation belong to the Hölder space $ C ^ \alpha ( \overline{D}\; ) $ for some $ \alpha > 0 $, that the coefficients $ a _ {i,j} ( x , t ) $ have derivatives $ \partial a _ {i,j} / \partial x _ {i} $ in $ C ^ \alpha ( \overline{D}\; ) $, $ i , j = 1 \dots n $, that $ f ( x , t ) $ belongs to $ C ^ \alpha ( \overline{D}\; ) $, that $ \phi $ and $ \psi $ are continuous on $ \overline \Omega \; _ {0} $ and $ \overline \Gamma \; $, respectively, and that $ \phi \mid _ {\partial \Omega } = \psi ( x , 0 ) $. For this it is sufficient that the boundary $ \partial \Omega $ of $ \Omega $ satisfies the following condition: For any point $ x ^ {0} \in \partial \Omega $ there is a closed sphere $ S $ having a unique point in common with $ \Omega $, namely, the point $ x ^ {0} $: $ S \cap \Omega = x ^ {0} $. Under certain conditions on the lateral surface (that it contains no characteristic points, i.e. points of contact with the planes $ t = \textrm{ const } $), an analogous statement also holds in the case of a non-cylindrical domain $ D $.

Existence theorems for the basic mixed problems for equation (1'}) also hold under other conditions on the given functions and the domain $ \Omega $. For example, in the case of the first mixed problem in a cylindrical domain $ D $, for the homogeneous heat equation with continuous functions $ \phi $ and $ \psi $ satisfying the compatibility condition $ \phi \mid _ {\partial \Omega } = \psi ( x , 0 ) $, a solution exists provided that $ \Omega $ is such that the Dirichlet problem for the Laplace equation is solvable in $ \Omega $( there is a classical solution) for an arbitrary continuous boundary function.

Let the coefficients $ a _ {ij} $, $ b _ {i} $ and $ c $ be measurable and bounded in $ D $, and let $ \sigma $ be measurable and bounded on $ \Gamma $. Further, let $ f \in L _ {2} ( D) $, $ \phi \in L _ {0} ( \Omega ) $ and, in the case of the first mixed problem, let $ \psi $ be the trace on $ \Gamma $ of some function from the Sobolev space $ W _ {2} ^ {1,0} ( D) $, while in the case of the third (or second) mixed problem, let $ \psi $ belong to $ L _ {2} ( \Gamma ) $.

A function $ u ( x , t ) $ belonging to $ W _ {2} ^ {1,0} ( D) $ and with trace on $ \Gamma $ equal to $ \psi $: $ u \mid _ \Gamma = \psi $, is called a generalized solution of the first mixed problem (1'}), (2'}), (4) if it satisfies the integral identity

$$ \int\limits _ { D } \left [ - u v _ {t} + \sum _ {i , j = 1 } ^ { n } a _ {i,j} u _ {x _ {i} } v _ {x _ {j} } - \left ( \sum _ { i= } 1 ^ { n } b _ {i} u _ {x _ {i} } + c u \right ) v \right ] d x d t = $$

$$ = \ \int\limits _ { D } f v d x d t + \int\limits _ {\Omega _ {0} } \phi v d x $$

for all $ v $ in the Sobolev space $ W _ {2} ^ {1} ( D) $ for which $ v \mid _ \Gamma = 0 $, $ v \mid _ {\Omega _ {T} } = 0 $.

A function $ u ( x , t ) $ belonging to $ W _ {2} ^ {1,0} ( D) $ is called a generalized solution of the third (second, if $ \sigma \equiv 0 $) mixed problem (1), (2), (6) if it satisfies the integral identity

$$ \int\limits _ { D } \left [ - u v _ {t} + \sum _ {i , j = 1 } ^ { n } \alpha _ {i,j} u _ {x _ {i} } v _ {x _ {j} } - \left ( \sum _ { i= } 1 ^ { n } b _ {i} u _ {x _ {i} } + c u \right ) v \right ] d x d t + $$

$$ + \int\limits _ \Gamma \sigma u v d S = $$

$$ = \ \int\limits _ { D } f v d x d t + \int\limits _ {\Omega _ {0} } \phi v d x + \int\limits _ \Gamma \psi v d S $$

for all $ v $ in $ W _ {2} ^ {1} $ such that $ v \mid _ {\Omega _ {T} } = 0 $.

A generalized solution of each of these problems exists and is unique; moreover, if $ f \in L _ {p} ( D) $ is continuous in $ D $, for sufficiently large $ p $, then it even satisfies a Hölder condition for some exponent $ \alpha > 0 $. By increasing the smoothness of the given functions and the boundary of the domain subject to the compatibility conditions, the smoothness of the generalized solution increases. For example, consider the heat equation with $ \phi \equiv 0 $ and $ \psi \equiv 0 $, and let $ \partial \Omega $ be a sufficiently smooth surface. Then the generalized solution of the first mixed problem belongs to $ W _ {2} ^ {2 ( s + 1 ) , s + 1 } ( D) $, provided that $ f \in W _ {2} ^ {2 s , s } ( D) $ and the compatibility conditions

$$ \tag{7 } f \mid _ {\partial \Omega _ {0} } = \ ( \Delta f + f _ {t} ) \mid _ {\partial \Omega _ {0} } = \dots = \ \sum _ {i = 0 } ^ { {s } - 1 } \Delta ^ {i} \left . \frac{\partial ^ {s - 1 - i } f }{\partial t ^ {s - 1 - i } } \right | _ {\partial \Omega _ {0} } $$

hold.

In particular, if $ f \in L _ {2} ( D) $, then the solution belongs to $ W _ {2} ^ {2,1} ( D) $ when $ ( x , t ) \in D $, it satisfies the heat equation and its trace on $ \Omega _ {0} $ is equal to zero. If $ f \in W _ {2} ^ {2s,s} $ for sufficiently large $ s $ and the compatibility conditions (7) hold, then by virtue of the imbedding theorems, the generalized solution is classical. An analogous statement holds for generalized solutions of the basic mixed problems for equation (1'}) when the coefficients are sufficiently smooth.

Let $ \Omega = \mathbf R ^ {n} $. The problem of finding a solution in the strip $ D = \mathbf R ^ {n} \times ( 0 , T ) $ for the parabolic system (1) satisfying the initial condition (2) on the characteristic $ \Omega _ {0} = \{ x \in \mathbf R ^ {n} , t = 0 \} $ is called the Cauchy problem for (1). A classical solution of the Cauchy problem (1), (2) is a vector function $ u ( x , t ) $ belonging to $ C ^ {2p,1} ( D) \cap C ( D \cup \Omega _ {0} ) $ and satisfying (1) in $ D $ and (2) on $ \Omega $. If the right-hand side $ f ( x , t ) $ belongs to the Hölder space $ C ^ \alpha ( \overline{D}\; ) $ for some $ \alpha > 0 $, and the coefficients are sufficiently smooth in $ \overline{D}\; $( they and their derivatives are bounded), then for any bounded continuous initial vector function $ \phi ( x) $ on $ \mathbf R ^ {n} $ there is a bounded solution of the Cauchy problem on $ D $, and this bounded solution is unique.

The condition of boundedness can be replaced by the condition of "not too rapid growth" . For example, the following is true for second-order equations. Let the coefficients of equation (1'}),

$$ a _ {i,j} ( x , t ) ,\ b _ {i} ( x , t ) ,\ c ( x , t ) \ \textrm{ and } \ \ \frac{\partial a _ {i,j} ( x , t ) }{\partial x _ {i} } $$

belong to the Hölder space $ C ^ \alpha ( \overline{D}\; ) $ for some $ \alpha > 0 $. Further, let $ \phi ( x) $ be continuous in $ \mathbf R ^ {n} $ and let $ f ( x , t ) $ be continuous in $ \overline{D}\; $, locally Hölder continuous in $ x $ uniformly for $ t \in [ 0 , T ] $( for some exponent $ \alpha > 0 $), and such that

$$ | \phi ( x) | \leq C e ^ {h | x | ^ {2} } ,\ \ x \in \mathbf R ^ {n} , $$

$$ | f ( x , t ) | \leq C e ^ {h | x | ^ {2} } ,\ ( x , t ) \in D . $$

Then for sufficiently small $ T $( depending on $ h $), there is a solution of the Cauchy problem (1'}), (2'}) in the strip $ D = \mathbf R ^ {n} \times ( 0 , T ) $. It can be written in the form

$$ u ( x , t ) = \int\limits _ {\mathbf R ^ {n} } \Gamma ( x , t ; \xi , 0 ) \phi ( \xi ) d \xi + $$

$$ + \int\limits _ { 0 } ^ { t } \int\limits _ {\mathbf R ^ {n} } \Gamma ( x , t ; \xi , \tau ) f ( \xi , \tau ) d \xi d \tau , $$

where $ \Gamma ( x , t ; \xi , \tau ) $ is a fundamental solution of (1'}), and satisfies the estimate

$$ \tag{8 } | u ( x , t ) | \leq C _ {1} e ^ {h | x | ^ {2} } $$

for some positive constants $ C _ {1} $ and $ k $. Condition (8) guarantees the uniqueness of the solution of the Cauchy problem.

In the case of an equation with constant coefficients it is possible to find a condition of type (8) on the growth of the solution that is necessary and sufficient for its uniqueness. For example, for a solution of the Cauchy problem for the heat equation to be unique in the class of functions satisfying the inequality

$$ | u ( x , t ) | \leq C e ^ {| x | h ( | x | ) } , $$

where $ h ( | x | ) $ is a positive continuous function on $ [ 0 , \infty ) $, it is necessary and sufficient that the integral $ \int _ {0} ^ \infty dr / h( r) $ diverges.

For parabolic equations one can also consider problems without initial conditions (the Fourier problem). For example, one can ask for the solution of the homogeneous heat equation in the cylinder $ D = \{ x \in \Omega, - \infty < t < + \infty \} $, where $ \Omega $ is a bounded domain with sufficiently smooth boundary $ \partial \Omega $, satisfying the boundary condition

$$ u ( x , t ) \mid _ {x \in \partial \Omega } = \psi ( x , t ) . $$

If $ \psi $ is continuous and bounded, then there is a bounded solution of the Fourier problem, and this is the unique bounded solution.

For parabolic equations and systems it is also possible to consider the first mixed problem in a non-cylindrical domain $ D $ in the case when the lateral surface contains characteristic points (points of contact with the planes $ t = \textrm{ const } $). In particular, it is possible to consider the Dirichlet problem, where boundary conditions are given on the entire boundary $ \partial \Omega $. Under specific conditions on the set of characteristic points and on the order of contact of the characteristic points of $ \partial \Omega $ with a characteristic plane, the Dirichlet problem has a unique solution (in the space $ W _ {2} ^ {B ^ {0} } $). For example, suppose (for the sake of simplicity) that $ D \subset \mathbf R ^ {2} $ is a strictly-convex domain and that the equation of the boundary in a neighbourhood of the upper characteristic point $ ( x ^ {0} , t ^ {0} ) $ has the form $ x - x ^ {0} = \phi _ {1} ( t) $ when $ x \leq x ^ {0} $, and $ x - x ^ {0} = \phi _ {2} ( t) $ when $ x \geq x ^ {0} $( $ t ^ {0} - \delta < t \leq t ^ {0} $). Then the divergence of both integrals

$$ \int\limits _ {t ^ {0} - \delta } ^ { {t ^ {0}} } | \phi _ {i} ( t) ^ {-} 2p | dt ,\ \ i = 1 , 2 , $$

guarantees the existence and uniqueness of a solution of the Dirichlet problem for a second-order parabolic equation. This condition is also necessary in this class of equations.

References

[1]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[2]	V.A. Il'in, "The solvability of mixed problems for hyperbolic and parabolic equations" Russian Math. Surveys , 15 : 2 (1960) pp. 85–142 Uspekhi Mat. Nauk , 15 : 2 (1960) pp. 97–154
[3]	A.M. Il'in, A.S. Kalashnikov, O.A. Oleinik, "Linear equations of the second order of parabolic type" Russian Math. Surveys , 17 : 3 (1962) pp. 1–143 Uspekhi Mat. Nauk , 17 : 3 (1962) pp. 3–146
[4]	S.N. Kruzhkov, "A priori bounds and some properties of solutions of elliptic and parabolic equations" Mat. Sb. , 65 : 4 (1964) pp. 522–570 (In Russian)
[5]	O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'tseva, "Linear and quasi-linear equations of parabolic type" , Amer. Math. Soc. (1968) (Translated from Russian)
[6]	O.A. Ladyzhenskaya, "The boundary value problems of mathematical physics" , Springer (1985) (Translated from Russian)
[7]	O.A. Ladyzhenskaya, "On uniqueness of a solution of the Cauchy problem for a linear parabolic equation" Mat. Sb. , 27 : 2 (1950) pp. 175–184 (In Russian)
[8]	V.P. Mikhailov, "The Dirichlet problem for a parabolic equation I" Mat. Sb. , 61 : 1 (1963) pp. 40–64 (In Russian)
[9]	V.P. Mikhailov, "The Dirichlet problem for a parabolic equation II" Mat. Sb. , 62 : 2 (1963) pp. 140–159 (In Russian)
[10]	J. Nash, "Continuity of solutions of parabolic and elliptic equations" Amer. J. Math. , 80 (1958) pp. 931–954
[11]	I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian)
[12]	I.G. Petrovskii, "On Cauchy's problem for systems of linear partial differential equations in a nonanalytic function domain" Bull. Moskov. Gos. Univ. Ser. A. , 1 : 7 (1938) pp. 1–72 (In Russian)
[13]	I.G. Petrovskii, "Zur ersten Randwertaufgabe der Wärmeleitungsgleichung" Comp. Math. , 1 (1934) pp. 383–419
[14]	S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian)
[15]	V.A. Solonnikov, "Boundary value problems of mathematical physics III" Proc. Steklov Inst. Math. , 83 (1965) Trudy Mat. Inst. Steklov , 83 (1965)
[16]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[17]	A.N. Tikhonov, Byull. Moskov. Univ. (A) , 1 : 9 (1938) pp. 1–43
[18]	A.N. Tikhonov, Mat. Sb. , 42 : 2 (1935) pp. 199–216
[19]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
[20]	S.D. Eidel'man, "Parabolic systems" , North-Holland (1969) (Translated from Russian)

Comments

In the current literature the initial boundary value problems (1'}), (2'}), (4)–(6) are not referred to as "mixed" . Sometimes expressions like Cauchy–Dirichlet or Cauchy–Neumann are used. Quite often, by a problem with Dirichlet data for a parabolic equation is meant a problem in which such data are prescribed on the parabolic boundary. Besides the first, second and third kind of boundary data, higher-order problems are also considered. For further comments and more references see Linear parabolic partial differential equation and system.