# Abstract parabolic differential equation

An equation of the form

(a1) |

where for each , is the infinitesimal generator of an analytic semi-group (cf. also Semi-group of operators; Strongly-continuous semi-group) in some Banach space . Hence, without loss of generality it is always assumed that

I) there exist an angle and a positive constant such that

i) , ;

ii) , , . The domain of is not necessarily dense. Various results on the solvability of the initial value problem for (a1) have been published. The main object is to construct the fundamental solution (cf. also Fundamental solution), which is an operator-valued function satisfying

The solution of (a1) satisfying the initial condition

(a2) |

if it exists, is given by

(a3) |

In parabolic cases, the fundamental solution usually satisfies the inequality

for some constant .

One of the most general result is due to P. Acquistapace and B. Terreni [a3], [a4]. Suppose that

II) there exist a constant and a set of real numbers with , , such that

Then the fundamental solution exists, and if satisfies (a2) and is Hölder continuous (i.e., for some , i.e.

(a4) |

cf. also Hölder condition), then the function (a3) is the unique solution of (a1), (a2) in the following sense: , for , , (a1) holds for and (a2) holds. A solution in this sense is usually called a classical solution. If, moreover, and , then , for , and (a1) is satisfied in . Such a solution is usually called a strict solution.

The following results on maximal regularity are well known.

### Time regularity.

Let and . Then

if and only if , where is the set of all functions such that (an interpolation space between and ; cf. also Interpolation of operators) for almost all and the norm of on is essentially bounded in .

### Space regularity.

Let and . Then

if and only if .

Hypothesis II) holds if the domain is independent of and is Hölder continuous, i.e. there exist constants and such that

Another main result by Acquistapace and Terreni is the following ([a2]):

III.i) is differentiable and there exist constants and such that

III.ii) there exist constants and such that

If is densely defined, this case reduces to the one in [a8]. Under the assumptions I), III) it can be shown that for and satisfying (a2) and (a4), a classical solution of (a1) exists and is unique. The solution is strict if, moreover, and

(a5) |

The following maximal regularity result holds: If and for , then the solution of (a1) belongs to if and only if the left-hand side of (a5) belongs to .

Another of general results is due to A. Yagi [a10], where the fundamental solution is constructed under the following assumptions:

IV) hypothesis III.i) is satisfied, and there exist constants and a non-empty set of indices satisfying such that

It is shown in [a4] that the above three results are independent of one another.

The above results are applied to initial-boundary value problems for parabolic partial differential equations:

where is an elliptic operator of order (cf. also Elliptic partial differential equation), are operators of order for each , and is a usually bounded open set in , , with smooth boundary . Under some algebraic assumptions on the operators , and smoothness hypotheses of the coefficients, it is shown in [a1] that the operator-valued function defined by

satisfies the assumptions I) and II) in the space if some negative constant is added to if necessary (this is not an essential restriction). The regularity of the coefficients here is Hölder continuity with some exponent. An analogous result holds for the operator defined in , by

There is also extensive literature on non-linear equations; see [a7] and [a9] for details. The following result on the quasi-linear partial differential equation

(a6) |

is due to H. Amann [a5], [a6]: For a given function , let be the solution of the linear problem

(a7) |

If the problem is extended to a larger space so that the domains of the extensions of are independent of , then, under a weak regularity hypothesis for on , the fundamental solution for the equation (a7) can be constructed, and a fixed-point theorem can be applied to the mapping to solve the equation (a6). The result has been applied to quasi-linear parabolic partial differential equations with quasi-linear boundary conditions.

#### References

[a1] | P. Acquistapace, "Evolution operators and strong solutions of abstract linear parabolic equations" Diff. and Integral Eq. , 1 (1988) pp. 433–457 |

[a2] | P. Acquistapace, B. Terreni, "Some existence and regularity results for abstract non-autonomous parabolic equations" J. Math. Anal. Appl. , 99 (1984) pp. 9–64 |

[a3] | P. Acquistapace, B. Terreni, "On fundamental solutions for abstract parabolic equations" A. Favini (ed.) E. Obrecht (ed.) , Differential Equations in Banach Spaces, Bologna, 1985 , Lecture Notes Math. , 1223 , Springer (1986) pp. 1–11 |

[a4] | P. Acquistapace, B. Terreni, "A unified approach to abstract linear non-autonomous parabolic equations" Rend. Sem. Univ. Padova , 78 (1987) pp. 47–107 |

[a5] | H. Amann, "Quasilinear parabolic systems under nonlinear boundary conditions" Arch. Rat. Mech. Anal. , 92 (1986) pp. 153–192 |

[a6] | H. Amann, "On abstract parabolic fundamental solutions" J. Math. Soc. Japan , 39 (1987) pp. 93–116 |

[a7] | H. Amann, "Linear and quasilinear parabolic problems I: Abstract linear theory" , Monogr. Math. , 89 , Birkhäuser (1995) |

[a8] | T. Kato, H. Tanabe, "On the abstract evolution equation" Osaka Math. J. , 14 (1962) pp. 107–133 |

[a9] | A. Lunardi, "Analytic semigroups and optimal regularity in parabolic problems" , Progr. Nonlinear Diff. Eqns. Appl. , 16 , Birkhäuser (1995) |

[a10] | A. Yagi, "On the abstract evolution equation of parablic type" Osaka J. Math. , 14 (1977) pp. 557–568 |

**How to Cite This Entry:**

Abstract parabolic differential equation. H. Tanabe (originator),

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