Asymptotic spatial behavior for the heat equation on noncompact regions

We consider the isotropic initial boundary value problem for the heat equation on open regions with noncom-pact boundary and construct differential inequalities for a generalized heat flow measure defined over a spherical cross section. Under suitable assumptions, integration of the differential inequality leads to spatial growth and decay rate estimates for mean-square cross-sectional measures of the time-weighted temperature spatial gradient. The estimates are then used to obtain similar results for the time-weighted temperature. In particular, when the base heat flow measure is positive, the time-weighted temperature becomes pointwise unbounded at large spatial distance.


INTRODUCTION
The initial boundary value problem is considered for the isotropic heat equation on open threedimensional regions with noncompact boundary subject to homogeneous initial data and zero temperature on appropriate portions of the boundary.Spatial asymptotic behavior, however, is not prescribed but is to be determined.The aim is to derive spatial growth and decay rate estimates for mean-square spherical cross-sectional integral measures of the time-weighted temperature gradient.Corresponding estimates for the time-weighted temperature follow with those for growth involving pointwise unbounded behavior.
Spherical cross-sectional surfaces are selected since, unlike plane cross sections, when the half-space and exterior regions are considered the a priori specification of asymptotic behavior is not required.
Although all results are believed to be new, that for unbounded generalized temperature is regarded of particular significance.
Our treatment is motivated by the procedure originally developed by Payne and Weinberger 1 and later by others including Ref. [2].We construct a differential inequality for the generalized heat flow across the cross section whose integration leads at each time instant to conditions for spatial growth and decay of the mean-square spatial gradient of the time-weighted temperature.The growth component, together with suitably adapted versions of the maximum principle and theorems due to Evans, 3 is then used to prove that the time-weighted temperature itself becomes pointwise unbounded at large spatial distances.The rate of increase is not established.On the other hand, the decay estimate leads to decay rates for the time-weighted crosssectional measures of both the temperature and its spatial gradient.Although the rates of growth and decay are algebraic, for a large subset of measures, the rates can be improved to become exponential.
Simple known examples illustrate essential features of the general procedure.
A similar treatment is applied by Quintanilla 4 to general parabolic equations for exponentially time-weighted measures related to the thermal energy.Spatial growth and decay rates are derived that are either at least or at most exponential.
Section 2 extends the notion of a geometrical structure introduced by Evans. 3 This preliminary section also states the initial boundary problem for both the temperature and its time-weighted generalizations.Section 3, for ease of reference, states without proof the usual maximum principle but is mainly devoted to a crucial generalization of an upper bound derived by Evans 3 in connection with regularity of the temperature on a bounded spatial region.
Section 4 discusses known explicit solutions to the one-dimensional isotropic heat equation on the semi-infinite line to illustrate the chief characteristics of the general treatment and also to demonstrate that our analysis is not merely formal.Section 5.1 introduces generalized heat flow measures defined on spherical cross-sections and derives various bounds.These together with bounds for the radial derivative obtained in Section 5.3 lead in Section 5.4 to a differential inequality whose integration in Section 6 under appropriate conditions yields growth estimates for cross-sectional measures of the spatial gradient of the time-weighted temperature valid at each positive time instant within the interval of existence.
The growth estimates combined with an extension of Evans' theorem are employed in Section 7 to prove by contradiction that the generalized temperature itself becomes asymptotically pointwise unbounded.The growth rate, however, remains open.Section 8 derives decay rate estimates for cross-sectional mean-square measures of both the generalized temperature and its spatial gradient.The method exploits the differential inequality established in Section 5. 4. Taken together, our results correspond to a Phragmén-Lindelöf principle.Section 9 is devoted to brief concluding remarks.
Vector and scalar quantities are not typographically distinguished.Subscripts, however, denote the respective Cartesian components of a vector.The comma notation indicates spatial partial differentiation, while a superposed dot denotes differentiation with respect to the time variable .The convention of summation over repeated suffixes is adopted, with Latin suffixes ranging over 1,2,3 and Greek suffixes over 1,2.

Basic geometry
Consider the open region Ω ⊆ IR 3 with noncompact boundary for which the half-space and exterior region are special cases.Suppose Ω is such that at least one semi-infinite ray drawn from a point on a finite part of the boundary lies entirely within Ω.Let this point be the origin  ∈ IR 3 of a rectangular Cartesian system of coordinates ( 1 ,  2 ,  3 ) whose positive  3 -axis coincides with the semi-infinite ray lying within Ω. Denote the spherical cross section Σ() by where (, ) is the ball of radius  > 0 centered at the point  ∈ IR 3 .Moreover, we set and let the noncompact boundary Ω of Ω, supposed Lipschitz smooth, be the union of disjoint parts Ω 1 and Ω 2 where Ω 1 ∩ Ω 2 = ∅, and Ω 1 satisfies for given  0 > 0. Without loss, take the origin  of the Cartesian coordinate system to be a point on Ω 1 , and define the time-independent spatial region Ω( 1 ,  2 ) by

Initial boundary value problem
The heat flux vector field   (, ) and temperature (, ) ≥ 0, assumed sufficiently smooth functions of position and time, are related on Ω( 0 , ∞) by where (0, ), 0 <  ≤ ∞, is the maximal time interval of existence of smooth solutions,  is a given positive bounded constant, and the components   () of the symmetric heat conduction tensor are supposed to be (piecewise) smooth and positive-definite in the sense that for vectors , there exists a positive constant  0 such that We also suppose that for positive bounded constant  1 , the heat conduction tensor satisfies The differential equation for the temperature, obtained from ( 5) and ( 6), is given by whose fundamental solution Φ(, ) for each  > 0 has the property: Remark 1. Treatments of the general heat conduction equation ( 9) are presented, for example, in the book [ 5 , p. 24] where derivations of the fundamental solution may also be found.Explicit expressions are not required here.
Specification of the heat conduction problem on Ω( 0 , ∞) is completed by assigning homogeneous initial and Dirichlet boundary conditions to be In Ω(0,  0 ) × [0, ), the heat conduction equation ( 9) is supplemented by prescribed heat sources, given mixed thermal boundary conditions on Ω 1 × [0, ), and homogeneous initial temperature in Ω(0,  0 ).Adjustment of these data yields the heat flux and temperature on Σ( 0 ) × [0, ) required to satisfy later assumptions.Data are not specified on the interior of Σ() for  0 <  ≤ ∞; in particular, spatial asymptotic behavior is to be determined.
The subsequent general discussion involves the integrals  () (, ) and not directly the temperature (, ).For convenience, however, the temperature is retained in Section 3 devoted to extensions of Evans' theorems and the maximum principle but obviously the results are valid also for  (𝑚) (, ).

Additional geometry
Various geometric structures based on those treated in Ref. [3] are now introduced.Let the spacetime region Ω ( 1 ,  2 ] be defined by where 0 ≤  1 <  2 ≤ .A corresponding notation is introduced for other space-time regions.The union (c.p., [ 3 , p. 51]), that explicitly excludes the region Ω( 0 , ∞) × { =  2 }, is called the parabolic boundary Introduce spherical polar coordinates (, , ) whose origin is that of the Cartesian coordinate system located at a point on Ω 1 .Here,  is the angle of inclination of the radius with the positive  3 -axis.For fixed , let  = (, 0, 0) be the point of intersection of the  3 -axis with Σ(), which is not necessarily at the center of the spherical cap Σ().
For given , define () by The cylinder (, ; ()) given by (19) relates to the fixed point  ∈ Σ().We next seek additional geometric structures that enable all points of Σ() to be included.
For this purpose, consider the great circle Δ(, ) on Σ() through the point  and lying in the plane that intercepts the  3 -axis at angle  to the  1 -axis.For fixed , , let the points  () (, ) = (, ,  (𝛼) ),  = 1, 2, lie on the great circle Δ(, ).The angles  (𝛼) are chosen such that the points  (𝛼) are on adjacent sides of .
In consequence, as the angle  varies in the interval [0, ], we have where the space-time region () is given by )
Remark 6 (Generalization).As stated in Remark 2, the time-weighted integrals  () (, ) defined by (13) satisfy the heat equation ( 14) and homogeneous initial and boundary conditions corresponding to (11) and (12).In consequence, Theorem 1 and Theorem 2 hold also for  (𝑚) (, ), which is exclusively used in the discussion from Section 5 onwards.It is shown, moreover, that additional results become possible that apparently are not available for (, ).

ILLUSTRATIVE EXAMPLES
We discuss two simple known examples that illustrate the growth and decay behavior predicted later and serve to confirm that the subsequent general discussion does not lack meaning.
Both examples involve the one-dimensional isotropic initial boundary value problem on the semi-infinite real line specified by where the prescribed function () satisfies (0) = 0, the solution (, ) is assumed sufficiently smooth, and (0, ) is the maximal interval of existence.

Spatial growth
An explicit solution to the initial boundary value problem ( 53)-( 55), attributed by F. John in Ref. [7, p. 211] to Tychonov 8 (see also Hellwig 9 ), exhibits unbounded spatial asymptotic growth and is given by where  () () denotes the -th derivative of () and  (0) () = ().Set for positive constant .However, unbounded spatial growth as  → ∞ of the infinite series (56) may be established irrespective of whether (, ) is positive or negative.The conclusion follows on first supposing that the infinite series (56) is positive and convergent for all  ∈ [0, ∞).Rearrangement gives (, ) = () +  2 (, ), where Let us further suppose that  is positive and bounded for all .Accordingly, by (60), (, ) becomes unbounded with , which contradicts the convergence assumption on (, ).Hence, assume that  remains positive but unbounded as  → ∞ to give a second contradiction.The assumption that  is negative and bounded implies that  is negative for sufficiently large  contrary to the assumption of positive .Consequently,  cannot be both positive and bounded.Suppose therefore that  is negative and bounded.A repeat of the argument leads to similar contradictions and to the conclusion that (, ) must be unbounded as  → ∞.That is, Example (56) also may be used to illustrate sufficient conditions later derived for spatial growth.These conditions are expressed in terms of the heat flow, which in the present case is given by (, ) ∶= ∫  0 (, ) , (, ) . ( Substitution from (53) and use of (55) show that so that the spatial derivative of  is nonnegative, characteristic of the general analysis for both growth and decay.In particular, the later sufficient general condition for growth requires the heat flow to be positive at some finite spatial distance, which by (64) implies the heat flow remains positive at all greater spatial distances.We sketch how this condition is satisfied for the heat flow measure (63) with () given by (57).
Observe that the derivatives of () may be expressed as the convergent infinite series where the nontensorial coefficients    are recursively obtained from Coefficients of special interest are 1 = (−1) Now rewrite the infinite series (65) as to conclude that for  = 0, 1, 2, … each derivative  () () remains positive for sufficiently small time and all .On the other hand, for sufficiently large time, the first term in (71) becomes dominant and according to (70),  () () alternates in sign with .Examination of the leading term in either (65) or (71) leads to the conclusion that for sufficiently large time there holds despite  (2) () being negative.Furthermore, the infinite series (56) may be expressed as the sum of two convergent series of positive and negative terms given by We appeal to (72) to obtain which is positive provided or 0 ≤  < √ 2. The infinite series (56) may be differentiated termwise for bounded , and the result decomposed into two convergent infinite series of positive and negative terms, which together with (72) leads to which is positive for 0 ≤  < √ 6.In consequence, the heat flow (63) is positive for 0 ≤  < √ 2 consistent with the general sufficient growth condition derived later.

Spatial decay
Decay behavior is subject to the asymptotic boundedness condition and is illustrated by a second solution to the initial boundary value problem ( 53)-( 55).Thus, consider (see, for example, Ref. [7, p. 222] and Ref. [3, p. 87]) It is immediate from (77) that (, ) is nonnegative on its domain of definition and possesses exponential spatial decay that implies the asymptotic behavior The general sufficient condition for spatial decay derived later is that the heat flow is negative at some finite distance.To check that the heat flow (63) satisfies this condition, we note from (77) that and consequently,  , (, ) is negative for  √ 2 <  0 <  at each time instant  ∈ [0, ).The heat flow (63) is therefore negative at a single sufficiently large  0 consistent with the general sufficient condition for spatial decay.Note that the heat flow must remain nonpositive for all  ≥  0 otherwise, as explained in the first example, sufficient conditions for unbounded spatial growth become satisfied in contradiction to (76).

DIFFERENTIAL INEQUALITIES FOR CROSS-SECTIONAL HEAT FLOW MEASURES
This and all later sections revert to the weighted temperature  () (, ) defined in (13) and assumed to satisfy (14) on the space-time region Ω (0,) together with initial and boundary conditions corresponding to (11) and (12).The analysis, conducted mainly in terms of the cross-sectional heat flow measure (, ), constructs differential inequalities whose integration lead to the required growth and decay rate estimates.

Generalized heat flow
For  = 0, 1, 2, … , define the cross-sectional heat flow measure (, ) by where  ≥ 0 for convenience is chosen to be nonnegative integer,  is the element of area on Σ(),   are the Cartesian components of the unit normal on Σ() in the increasing radial direction, the In what follows, we frequently distinguish the cases  ≥ 1 and  = 0.

𝛾 ≥ 1
We now consider bounds derived from (84) when  ≥ 1.Take  = 1 and when  is infinite let  ∈ [0,  1 ] for some  1 < ∞ to have the simple inequality We conclude from (83) that where On the other hand, the choice , in (84) yields the estimate which, subject to the condition Condition (90) for given  ≥ 1 and  1 < ∞ restricts the values of the coefficients ,  1 for which (91) is valid.

Alternative expression for the heat flow
The heat flow (, ) is next expressed in terms of space-time integrals.Different expressions are obtained for  ≥ 1 and  = 0.

Bounds for the radial derivative of the heat flow
As previously, the cases  ≥ 1 and  = 0 are separately discussed.
Proposition 1 (Poincaré-Wirtinger).Let Ψ() be a differential function defined on Σ() that vanishes on the boundary Σ().Then Ψ satisfies the inequality where ∇  denotes the tangential derivative on Σ().The positive constant  is bounded by where  is the smallest value of  such that   (cos ) = 0,   (.) specifies the Legendre polynomial of order , and 2 is the vertical angle of the circular cone containing Ω( 0 , ∞).In particular,  = 1∕2 both for the half-space and exterior regions provided that on such exterior regions Ψ is normalized by We deduce from (106) that

Differential inequalities
A differential inequality for the heat flow (, ) is derived for the general case, with additional results derived in Section 5.4.2 for  ≥ 1.

𝛾 ≥ 1
Now let  ≥ 1.A second differential inequality obtained from ( 91) and ( 103) subject to condition (90) is given by Otherwise, we conclude from (87) and (104) that where and  and  are defined in (105) and (88).
The respective differential inequalities (117)-(119) generate quantitatively different estimates as shown in subsequent sections.
But (, ) is nondecreasing with respect to  by inequality (102) so that and consequently differential inequality (117) may be written as where the positive constant  is defined in (116).Integration leads to the algebraic increasing lower bound: ( 0 , ) which combined with (115) yields ( 0 , ) where Observe that for  ≥ 1, the quadratic forms  1 , 1 for the mean-square cross-sectional spatial gradient of the (time-weighted) temperature remain strictly positive.They algebraicly increase with respect to  when  > 1, which, however, restricts the constant  appearing in the Poincaré-Wirtinger inequality (106).

𝜸 ≥ 𝟏
Exponential growth occurs when  ≥ 1 subject to condition (90).By virtue of (122), differential inequality (118) reduces to (, ) ≤  , (, ), which for fixed  ∈ (0, ) integrates to give the growth rate estimate An improved estimate, based on inequality (119), is given by where  is defined in (120).Consequently, (, ) exhibits at least exponential spatial growth without restriction on either , , or  but only for  ≥ 1.Furthermore, on noting (115), we may derive from (129) and (130) estimates in terms of  1 (, ) or  (0) 1 introduced in (127) and (128).For example, we have the growth rate bound It is of interest to also express the growth estimates in terms of the energy functions (95) and (98).On using (96) and (97), we obtain ( 0 , ) and consequently conclude that ( 0 , ∞; ) and  (0) ( 0 , ∞; ) do not exist when (, ) becomes positive at any distance  ≥  0 .Unbounded asymptotic behavior is alternatively represented on application of l'Hôpital's theorem to (132), which for  ≥ 1 gives: where 0 <  0 < ∞,  1 (, ) is defined in (127), and Similar conclusions hold for  = 0 with  1 (, ) replaced by Although, for example, (125) and (131) establish growth of the cross-sectional gradient measure  1 (, ), it is not possible to prove from these estimates or from the asymptotic behavior (134) whether various cross-sectional measures of the time-weighted temperature become asymptotically unbounded and in what sense.The analysis required to prove such behavior is described in the next section where it is shown that  () (, ) subject to (121) and other appropriate conditions must become unbounded for sufficiently large .The actual rate of growth is not determined.

SPATIAL ASYMPTOTIC GROWTH OF THE TIME-WEIGHTED TEMPERATURE
The argument is developed for  ≥ 1, with reference to  = 0 as required.Consequently, in this section we employ the spatial growth estimates (125), and (131) for the mean-square gradient  1 (, ) to prove that subject to (121) the time-weighted temperature  () (, ),  = 0, 1, 2 …, becomes unbounded for sufficiently large  at each time  ∈ (0, ).
We proceed to establish a contradiction and accordingly suppose that  () (, ) remains bounded on Ω [0,] and that Theorem 2 applies.Recall that  () (, ) achieves its maximum either on Σ( 0 ) × (0, ] or on Σ(∞) × (0, ].The maximum cannot occur at an interior point since  (𝑚) would then be constant in contradiction to the gradient growth estimates for  1 and  and that  >  (1) ≥  (2) , we obtain by addition of ( 140 3 (2) )2 for some bounded constant D10 .The various increasing lower bounds (125) (for  ≥ 0 and  > 3), or (131) (for  ≥ 1 but unrestricted ), may then be combined with inequality (142) to show that max () becomes unbounded as  → ∞, contrary to the assumption that  () (, ) remains bounded.Under these conditions,  () (, ) must be unbounded at each  ∈ (0, ) as  → ∞.The rate of growth, however, has not been established, and it cannot be concluded that the class of unbounded solutions satisfying the increasing upper bound (28) is necessarily empty.

SPATIAL DECAY ESTIMATES
Let us suppose that the energies ( 0 , ∞; ) and  (0) ( 0 , ∞; ) are bounded and in particular that Corresponding inequalities obtained from (118) and (119) lead to exponential decay but only for  ≥ 1.