On existence and uniqueness for a new class of nonlinear partial differential equations using compactness methods and differential difference schemes
Authors: Dushane Theodore E. 
Published: July 2010

We prove existence and uniqueness results for the following Cauchy problem in the half plane
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to 0 colon u Subscript t Baseline plus left-parenthesis f left-parenthesis u right-parenthesis right-parenthesis Subscript x Baseline plus u Subscript x x x Baseline equals g 1 left-parenthesis u right-parenthesis u Subscript x x Baseline plus g 2 left-parenthesis u right-parenthesis left-parenthesis u Subscript x Baseline right-parenthesis squared plus p left-parenthesis u right-parenthesis comma u left-parenthesis x comma 0 right-parenthesis equals u 0 left-parenthesis x right-parenthesis">

t
≥
0
:

u
t

+

(
f
(
u
)

)
x

+

u

x
x
x

=

g
1

(
u
)

u

x
x

+

g
2

(
u
)

(

u
x

)
2

+
p
(
u
)
,
u
(
x
,
0
)
=

u
0

(
x
)

t \geq 0:{u_t} + {(f(u))_x} + {u_{xxx}} = {g_1}(u){u_{xx}} + {g_2}(u){({u_x})^2} + p(u),u(x,0) = {u_0}(x)

, where

u
=
u
(
x
,
t
)

u = u(x,t)

and the subscripts indicate partial derivatives. We require that f,

g
1

{g_1}

,

g
2

{g_2}

, and p be sufficiently smooth and satisfy

f
′

(
u
)
≥
0
,

∫
0
u

f
(
v
)

d
v
≥
0

f’(u) \geq 0,\smallint _0^uf(v)\;dv \geq 0

, and other similar sign conditions on

g
1

{g_1}

,

g
2

{g_2}

, and p. Our hypotheses allow for exponential growth of f,

g
1

{g_1}

,

g
2

{g_2}

, and p so long as the sign conditions are satisfied and include the special cases
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis u right-parenthesis equals u Superscript 2 n plus 1 Baseline comma g 1 left-parenthesis u right-parenthesis equals u Superscript 2 m Baseline comma g 2 left-parenthesis u right-parenthesis equals minus u Superscript 2 r plus 1">

f
(
u
)
=

u

2
n
+
1

,

g
1

(
u
)
=

u

2
m

,

g
2

(
u
)
=
−

u

2
r
+
1

f(u) = {u^{2n + 1}},{g_1}(u) = {u^{2m}},{g_2}(u) = - {u^{2r + 1}}

, and

p
(
u
)
=
−

u

2
s
+
1

p(u) = - {u^{2s + 1}}

, for n, m, r, and s nonnegative integers. To obtain a global solution in time, we perturb the equation by

−
ϵ
(

u

x
x
x
x

−

(
f
(
u
)

)

x
x

)

- \epsilon ({u_{xxxx}} - {(f(u))_{xx}})

. The perturbed equation is solved locally (in time) and this solution is extended to a global solution by means of a priori estimates on the

H
s

{H^s}

(of space) norms of the local solution. These estimates require the use of new nonlinear functionals. We then obtain the solution to the original equation as a limit of solutions to the perturbed equation as

ϵ
\epsilon

tends to zero using the standard techniques. For the related periodic problem, for which we require

u
(
x
+
2
π
,
t
)
=
u
(
x
,
t
)

u(x + 2\pi ,t) = u(x,t)

for all

t
≥
0

t \geq 0

, we also obtain existence and uniqueness results. We prove existence for this problem via similar techniques to the nonperiodic case. We then consider differential difference schemes for the periodic initial value problem and show that we may obtain the solution as the limit of solutions to an appropriate scheme.