Let
q
q
,
a
a
,
T
T
, and
b
b
be any real numbers such that
q
≥
0
q \ge 0
,
a
>
0
a > 0
,
T
>
0
T > 0
, and
0
>
b
>
1
0 > b > 1
. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at
b
b
:
\[
x
q
u
t
−
u
x
x
=
a
2
δ
(
x
−
b
)
f
(
u
(
x
,
t
)
)
i
n
(
0
,
1
)
×
(
0
,
T
]
,
{x^q}{u_t} - {u_{xx}} = {a^2}\delta \left ( x - b \right )f\left ( u\left ( x, t \right ) \right ) in \left ( 0, 1 \right ) \times \left ( 0, T \right ],
\]
\[
u
(
x
,
0
)
=
0
o
n
[
0
,
1
]
,
u
(
0
,
t
)
=
u
(
1
,
t
)
=
0
f
o
r
0
>
t
≤
T
,
u\left ( x, 0 \right ) = 0 on \left [ 0, 1 \right ], u\left ( 0, t \right ) = u\left ( 1, t \right ) = 0 for \; 0 > t \le T,
\]
where
δ
(
x
)
\delta \left ( x \right )
is the Dirac delta function,
f
f
is a given function such that
lim
u
→
c
−
f
(
u
)
=
∞
{\lim _{u \to {c^ - }}}f\left ( u \right ) = \infty
for some positive constant
c
c
, and
f
(
u
)
f\left ( u \right )
and
f
′
(
u
)
f’\left ( u \right )
are positive for
0
≤
u
>
c
0 \le u > c
. It is shown that the problem has a unique continuous solution
u
u
before
m
a
x
{
u
(
x
,
t
)
:
0
≤
x
≤
1
}
max\left \{ {u\left ( x, t \right ) : 0 \le x \le 1} \right \}
reaches
c
−
{c^ - }
,
u
u
is a strictly increasing function of
t
t
for
0
>
x
>
1
0 > x > 1
, and if
m
a
x
{
u
(
x
,
t
)
:
0
≤
x
≤
1
}
max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}
reaches
c
−
{c^ - }
, then
u
u
attains the value
c
c
only at the point
b
b
. The problem is shown to have a unique
a
∗
{a^*}
such that a unique global solution
u
u
exists for
a
≤
a
∗
a \le {a^*}
, and
m
a
x
{
u
(
x
,
t
)
:
0
≤
x
≤
1
}
max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}
reaches
c
−
{c^ - }
in a finite time for
a
>
a
∗
a > {a^*}
; this
a
∗
{a^*}
is the same as that for
q
=
0
q = 0
. A formula for computing
a
∗
{a^*}
is given, and no quenching in infinite time is deduced.