On nonexistence of Baras–Goldstein type for higher-order parabolic equations with singular potentials
Authors: Galaktionov V. , Kamotski I. 
Published: April 2010

The celebrated result by Baras and Goldstein (1984) established that the heat equation with singular inverse square potential in a smooth bounded domain

Ω
⊂

R

N

\Omega \subset \mathbb {R}^N

,

N
≥
3

N \ge 3

, such that

0
∈
Ω

0 \in \Omega

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals normal upper Delta u plus StartFraction c Over StartAbsoluteValue x EndAbsoluteValue squared EndFraction u in normal upper Omega times left-parenthesis 0 comma upper T right-parenthesis comma u vertical-bar Subscript partial-differential normal upper Omega Baseline equals 0 comma">

u
t

=
Δ
u
+

c

|

x

|

2

u

in

Ω
×
(
0
,
T
)
,

u

|

∂
Ω

=
0
,

u_t= \Delta u + \frac c{|x|^2} u \;\; \text {in} \;\; \Omega \times (0,T), \;\; u \big |_{\partial \Omega }=0,

\]
in the supercritical range
\[

c
>

c

H
a
r
d
y

(
1
)
=

(

N
−
2

2

)

2

,

c > c_{\mathrm {Hardy}}(1) = \big (\frac {N-2}2\big )^2,

\]
does not have a solution for any nontrivial

L
1

L^1

initial data

u
0

(
x
)
≥
0

u_0(x) \ge 0

in

Ω
\Omega

or for a positive measure. Namely, it was proved that a regular approximation of a possible solution by a sequence

{

u
n

(
x
,
t
)
}

\{u_n(x,t)\}

of classical solutions of uniformly parabolic equations with bounded truncated potentials given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V left-parenthesis x right-parenthesis equals StartFraction c Over StartAbsoluteValue x EndAbsoluteValue squared EndFraction right-arrow from bar upper V Subscript n Baseline left-parenthesis x right-parenthesis equals min left-brace right-brace comma c vertical-bar vertical-bar x 2 comma n left-parenthesis n greater-than-or-equal-to 1 right-parenthesis">

V
(
x
)
=

c

|

x

|

2

↦

V
n

(
x
)
=
min

{

c

|

x

|

2

,

n

}

(
n
≥
1
)

V(x) = \frac c{|x|^2} \mapsto V_n(x)=\min \big \{ \frac c{|x|^2}, \, n \big \} \,\,\, (n \ge 1)

\]
diverges, and, as

n
→
∞

n \to \infty

,
\[

u
n

(
x
,
t
)
→
+
∞

in

Ω
×
(
0
,
T
)
.

u_n(x,t) \to +\infty \quad \mbox {in} \quad \Omega \times (0,T).

\]
In the present paper, we reveal the connection of this “very singular” evolution with a spectrum of some “limiting” operator. The proposed approach allows us to consider more general higher-order operators (for which Hardy’s inequalities were known since Rellich, 1954) and initial data that are not necessarily positive. In particular it is established that, under some natural hypothesis, the divergence result is valid for any

2
m

2m

th-order parabolic equation with singular potential
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals minus left-parenthesis negative normal upper Delta right-parenthesis Superscript m Baseline u plus StartFraction c Over StartAbsoluteValue x EndAbsoluteValue Superscript 2 m Baseline EndFraction u normal i normal n normal upper Omega times left-parenthesis 0 comma upper T right-parenthesis comma normal w normal h normal e normal r normal e c greater-than c Subscript normal upper H Baseline left-parenthesis m right-parenthesis comma semicolon m greater-than-or-equal-to 1 comma">

u
t

=
−
(
−
Δ

)
m

u
+

c

|

x

|

2
m

u

i
n

Ω
×
(
0
,
T
)
,

w
h
e
r
e

c
>

c

H

(
m
)
,

;
m
≥
1
,

u_t = -(-\Delta )^m u + \frac c{|x|^{2m}}\, u \;\; \mathrm {in} \;\; \Omega \times (0,T), \;\; \mathrm {where} \;\; c>c_{\mathrm {H}}(m), \,; m \ge 1,

\]
with zero Dirichlet conditions on

∂
Ω

\partial \Omega

and for a wide class of initial data. In particular, typically, the divergence holds for any data satisfying
\[

u
0

(
x
)

is continuous at 

x
=
0

 and 

u
0

(
0
)
>
0

.

u_0(x) \;\; \text {is continuous at $x=0$ and $u_0(0)>0$}.

\]
Similar nonexistence (i.e., divergence as

ε
→
0

\varepsilon \to 0

) results are also derived for time-dependent potentials

ε

−
2
m

q
(

x
ε

,

t

ε

2
m

)

\varepsilon ^{-2m}q(\frac {x}{\varepsilon }, \frac {t}{\varepsilon ^{2m}})

and nonlinear reaction terms

|

u

|

p

ε

2
m

+

|

x

|

2
m

\frac {|u|^p}{\varepsilon ^{2m}+|x|^{2m}}

with

p
>
1

p>1

. Applications to other, linear and semilinear, Schrödinger and wave PDEs are discussed.