View source for Linear parabolic partial differential equation and system

A partial differential equation (system) of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593803.png" /> are natural numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593804.png" /> is an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593806.png" />, considered in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593807.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593808.png" />. The system (1) is said to be (Petrovskii) parabolic at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l0593809.png" /> if the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938011.png" />, of the polynomial (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938012.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938013.png" /></td> </tr></table>

satisfy the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938015.png" /> with imaginary unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938017.png" /> is the Kronecker symbol.

The system (1) is parabolic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938018.png" /> if the inequality (2) is satisfied for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938019.png" />, and uniformly parabolic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938020.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938021.png" /></td> </tr></table>

for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938022.png" />.

For the case of a second-order equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

one can give another definition of parabolicity. For a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938024.png" /> there is an affine transformation that takes (3) to the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938025.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938027.png" />. Equation (3) is parabolic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938028.png" /> if one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938029.png" /> (say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938030.png" />) is equal to zero, the other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938032.png" />, have the same sign and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938033.png" />. Equation (3) is parabolic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938034.png" /> if it is parabolic at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938035.png" />. If the coefficients of an equation (3) that is parabolic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938036.png" /> are sufficiently smooth, then in a neighbourhood of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938037.png" /> by a non-singular change of variables it can be reduced to the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

with a positive-definite form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938039.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938041.png" /> that satisfies (4) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938043.png" />, and at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938044.png" /> satisfies the initial condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938045.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938046.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938047.png" /> is a region in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938048.png" />. It is required to find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938049.png" /> satisfying the initial condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938050.png" /></td> </tr></table>

and the boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938052.png" /></td> </tr></table>

or the third

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938053.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938055.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938056.png" />. Also, for the Cauchy–Dirichlet problem, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938057.png" /> is unbounded for the boundary value problems, it is also required that the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938058.png" /> is bounded as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938059.png" /> (or, more generally, that the growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938060.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938062.png" /> 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|maximum principle]]. Suppose that the coefficients of (4) are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938063.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938064.png" /> is bounded; let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938065.png" /></td> </tr></table>

and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938066.png" /></td> </tr></table>

Then for any solution

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938067.png" /></td> </tr></table>

of equation (4) the estimate

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938068.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938069.png" /></td> </tr></table>

satisfying (5) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938070.png" />, and, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938071.png" /> bounded and continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938072.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938073.png" /></td> </tr></table>

uniformly on compact subsets of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938074.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938075.png" /> one obtains the solution

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

of the Cauchy–Dirichlet problem. All values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938077.png" /> influence the value of the solution at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938079.png" />. 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.D. Eidel'man,   "Parabolic systems" , North-Holland  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR></table>



====Comments====
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|characteristic]]; cf. also [[Cauchy characteristic problem|Cauchy characteristic problem]]; [[Cauchy 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059380/l05938080.png" />, 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(2)|Neumann problem]]) or in a different direction (the [[Third boundary value problem|third boundary value problem]]).

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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.A. Baderko,   "Solution of a heat conduction problem for concentrated heat capacities by the method of parabolic potentials"  ''Differential Eq.'' , '''8'''  (1972)  pp. 940–947  ''Differentsial. Uravn.'' , '''8'''  (1972)  pp. 1225–1234</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.R. Cannon,   "The one-dimensional heat equation" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Ciliberto,   "Formule di maggiorazione e teoremi di esistenza per le soluzioni delle equazioni paraboliche in due variabili"  ''Richerche di Mat.'' , '''3'''  (1954)  pp. 1234–1249</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Gevrey,   "Oeuvres" , C.N.R.S.  (1970)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Ughi,   "Stime a priori la soluzioni di problemi al contorno di quarto e quinto tipo per una equazione parabolica lineare"  ''Atti Sem. Mat. Fis. Univ. Modena'' , '''26'''  (1977)  pp. 304–328</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Flato,   "Deformation view of physical theories"  ''Czechoslovak J. Phys.'' , '''B32'''  (1982)  pp. 472–475</TD></TR></table>