Linear parabolic partial differential equation and system

A partial differential equation (system) of the form

$$ \tag{1 } \frac{\partial ^ {k _ {i} } u _ {i} }{\partial t ^ {k _ {i} } } = \sum_{j=1}^ { N } \sum _ {p s _ {0} + | s| \leq p k _ {j} } A _ {s _ {0} s } ^ {ij} ( x , t ) \frac{\partial ^ {s _ {0} } }{\partial t ^ {s _ {0} } } \frac{\partial ^ {s} }{\partial x ^ {s} } u _ {j} + f _ {i} ( x , t ) , $$

where $ 1 \leq i \leq N $, $ k _ {0} \dots k _ {N} $ are natural numbers, $ p $ is an integer, $ s = ( s _ {1} \dots s _ {n} ) $, $ | s | = s _ {1} + \dots + s _ {n} $, considered in a region $ D $ of the variables $ ( x , t ) = ( x _ {1} \dots x _ {n} , t ) $. The system (1) is said to be (Petrovskii) parabolic at a point $ ( x ^ {0} , t ^ {0} ) \in D $ if the roots $ \lambda _ {m} ( \xi , x , t ) $, $ 1 \leq m \leq k _ {1} + \dots + k _ {N} $, of the polynomial (in $ \lambda $)

$$ \mathop{\rm det} \ \left ( \sum _ {p s _ {0} + | s| = p k _ {j} } A _ {s _ {0} s } ^ {ij } \lambda ^ {s _ {0} } ( i \xi ) ^ {s} - \delta _ {ij} \lambda ^ {k _ {i} } \right ) $$

satisfy the inequality

$$ \tag{2 } \sup _ {\begin{array}{c} m \\ | \xi | = 1 \end{array} } \ \mathop{\rm Re} \lambda _ {m} ( \xi , x ^ {0} , t ^ {0} ) < 0 . $$

Here $ ( i \xi ) ^ {s} = ( i \xi _ {i} ) ^ {s _ {1} } \dots ( i \xi _ {n} ) ^ {s _ {n} } $ with imaginary unit $ i $, and $ \delta _ {ij} $ is the Kronecker symbol.

The system (1) is parabolic in $ D $ if the inequality (2) is satisfied for all $ ( x , t ) \in D $, and uniformly parabolic in $ D $ if

$$ \sup _ { \begin{array}{c} m \\ | \xi | = 1 \\ ( x , t ) \in D \end{array} } \ \mathop{\rm Re} \lambda _ {m} ( \xi , x , t ) < - \delta $$

for some constant $ \delta > 0 $.

For the case of a second-order equation

$$ \tag{3 } \sum _ {i , j = 0 } c _ {ij} u _ {x _ {i} x _ {j} } + \sum_{i=0}^ { n } c _ {i} u _ {x _ {i} } + c u = h $$

one can give another definition of parabolicity. For a given point $ x ^ {0} = ( x _ {0} ^ {0} \dots x _ {n} ^ {0} ) $ there is an affine transformation that takes (3) to the form

$$ \sum _ {i , j = 0 } ^ { n } b _ {ij} v _ {x _ {i} x _ {j} } + \sum_{i=0}^ { n } b _ {i} v _ {x _ {i} } + b v = g $$

with $ b _ {ij} ( x ^ {0} ) = 0 $ for $ i \neq j $. Equation (3) is parabolic at $ x ^ {0} $ if one of the $ b _ {ii} ( x ^ {0} ) $( say $ b _ {00} ( x ^ {0} ) $) is equal to zero, the other $ b _ {ii} ( x ^ {0} ) \neq 0 $, $ i > 0 $, have the same sign and $ b _ {0} ( x ^ {0} ) \neq 0 $. Equation (3) is parabolic in $ D $ if it is parabolic at every point of $ D $. If the coefficients of an equation (3) that is parabolic in $ D $ are sufficiently smooth, then in a neighbourhood of any point $ x ^ {0} \in D $ by a non-singular change of variables it can be reduced to the form

$$ \tag{4 } u _ {t} - \sum _ {i , j = 1 } ^ { n } a _ {ij} u _ {x _ {i} x _ {j} } + \sum_{i=1}^ { n } a _ {i} u _ {x _ {i} } + a u = f $$

with a positive-definite form $ \sum a _ {ij} \xi _ {i} \xi _ {j} $.

A typical representative of a parabolic equation is the thermal-conductance equation (or heat equation)

$$ \tag{5 } u _ {t} - \sum_{j=1}^ { n } u _ {x _ {i} x _ {i} } = 0 , $$

the main properties of which are preserved for general parabolic equations.

The following problems are fundamental for equation (4).

The Cauchy–Dirichlet problem: To find a function $ u ( x , t ) $ that satisfies (4) for $ x \in \mathbf R ^ {n} $, $ t > 0 $, and at $ t = 0 $ satisfies the initial condition

$$ u \mid_{t=0} = \phi ( x) ,\ \ x \in \mathbf R ^ {n} . $$

The first boundary value problem, in which (4) is specified in a cylinder

$$ \overline{Q}\; _ {T} = \overline \Omega \; \times [ 0 , T ] , $$

where $ \Omega $ is a region in $ \mathbf R ^ {n} $. It is required to find a function $ u $ satisfying the initial condition

$$ u \mid_{t=0} = \phi ( x) ,\ \ x \in \Omega , $$

and the boundary condition

$$ \tag{6 } u \mid _ {\begin{array} {c} x \in \partial \Omega \\ 0 \leq t \leq T \end{array} } = \ \psi ( x , t ) . $$

The second and third boundary value problems differ from the first only in condition (6), which is replaced by the second boundary value condition

$$ \left . \frac{\partial u }{\partial N } \right | _ {\begin{array} {c} x \in \partial \Omega \\ 0 \leq t \leq T \end{array} } \equiv \sum _ {i , j = 1 } ^ { n } a _ {ij} u _ {x _ {i} } \nu _ {i} = \ \psi ( x , t ) , $$

or the third

$$ \left ( \frac{\partial u }{\partial N } + \sigma u \right ) _ {\begin{array} {c} x \in \partial \Omega \\ a \leq t \leq T \end{array} } = \ \psi ( x , t ) , $$

where $ \nu _ {i} $, $ 1 \leq i \leq n $, are the components of the outward normal.

The classical formulation of these problems requires that the solution is continuous in the closed domain, that the derivatives with respect to the spatial variables up to the second order are continuous inside the domain, and in the case of the second and third boundary value problems that the first derivatives are continuous up to the lateral surface of the cylinder $ \Omega $. Also, for the Cauchy–Dirichlet problem, or if $ \Omega $ is unbounded for the boundary value problems, it is also required that the solution $ u $ is bounded as $ | x | \rightarrow \infty $( or, more generally, that the growth of $ | u | $ is specified in a suitable way).

Suppose that equation (4) is uniformly parabolic and that the coefficients of the equation, the initial and boundary conditions and the boundary of the domain are sufficiently smooth, and that for unbounded domains appropriate growth conditions are satisfied by the initial data. Then the solutions of the Cauchy–Dirichlet problem and the first boundary value problem exist and are unique. If $ a \leq 0 $, $ \sigma > 0 $ and if the necessary compatibility conditions are satisfied, then a similar result also holds for the second and third boundary value problems.

Uniqueness in these problems follows from the maximum principle. Suppose that the coefficients of (4) are continuous in $ \overline{Q}\; _ {T} $ and that $ \Omega $ is bounded; let

$$ \Gamma = \partial Q _ {T} \setminus \{ {( x , t ) } : {x \in \Omega , t = T } \} $$

and let

$$ M = \max _ {\overline{Q}\; _ {T} } a ,\ \ N = \max _ {\overline{Q}\; _ {T} } | f | . $$

Then for any solution

$$ u \in C ( \overline{Q}\; _ {T} ) \cap C ^ {2} ( \overline{Q}\; _ {T} \setminus \Gamma ) $$

of equation (4) the estimate

$$ | u ( x , t ) | \leq e ^ {mt} \left ( N t + \max _ \Gamma | u | \right ) ,\ \ ( x , t ) \in Q _ {T} , $$

holds. The maximum principle can also be extended to the case of unbounded domains. In addition, for parabolic equations an analogue of the Zaremba–Giraud principle holds, concerning the sign of the inclined derivative at an extremum, which is well known in the theory of elliptic equations.

In the theory of parabolic equations an important role is played by fundamental solutions. In the case of the heat equation (5) such is the function

$$ w ( x , t , \xi , \tau ) = \ \frac{1}{2 \sqrt {\pi ( t - \tau ) } } e ^ { ( x - \xi ) ^ {2} / 4 ( t - \tau ) } , $$

satisfying (5) for $ t > \tau $, and, for any function $ \phi ( x) $ bounded and continuous in $ \mathbf R ^ {n} $,

$$ \lim\limits _ {t \rightarrow \tau + 0 } \ \int\limits _ {\mathbf R ^ {n} } w ( x , t , \xi , \tau ) \phi ( \xi ) d \xi = \phi ( x) $$

uniformly on compact subsets of points $ x \in \mathbf R ^ {n} $. In particular, for $ \tau = 0 $ one obtains the solution

$$ \tag{7 } u ( x , t ) = \int\limits _ {\mathbf R ^ {n} } w ( x , t , \xi , 0 ) \phi ( \xi ) d \xi $$

of the Cauchy–Dirichlet problem. All values of the function $ \phi ( x) $ influence the value of the solution at a point $ ( x , t ) $, $ t > 0 $. This is an expression of the fact that perturbations of the Cauchy–Dirichlet problem are propagated with infinite speed. This is the essential difference between parabolic equations and hyperbolic equations, where the speed of propagation of perturbations is finite.

Fundamental solutions can also be constructed for general parabolic equations and systems under very general assumptions about the smoothness of the coefficients.

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It must be stressed that the name "Cauchy–Dirichlet problem" is usually attached to the first boundary value problem. The initial value problem in the whole space is called the Cauchy or the characteristic Cauchy problem (because the data are prescribed on a characteristic; cf. also Cauchy characteristic problem; Cauchy problem).

Also, the distinction made in the text between the second and the third boundary value problem is not the one usually found: In the Western literature the two problems are discriminated not by the presence of the term $ \sigma u $, but by the fact that the directional derivative appearing in the boundary condition is in the direction of the conormal (the second boundary value or Neumann problem) or in a different direction (the third boundary value problem).

The fourth and fifth boundary value problems are also of some importance (see [a1], [a5]).

A basic role in the theory of linear parabolic equations is played by estimates of Schauder type, which were obtained in [a3]. The classical works by M. Gevrey [a4] are a milestone in the theory of parabolic equations.

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