This paper deals with the initial value problem of the type
|
\frac¶u(t,x) ¶t = Lu(t,x), u(0,x) = u0(x)\frac{\partial u(t,x)} {\partial t} = {\mathcal{L}}u(t,x), \quad u(0,x) = u_{0}(x)
|
(0.1) |
in Banach spaces with
Lp-norm, where
t is the time,
u0 is a monogenic function and the operator
L{\mathcal{L}} is of the form
|
Lu(t,x) : = åA,B,i C(A)B,i(t,x) \frac¶uB(t,x) ¶xi eA{\mathcal{L}}u(t,x) := \sum_{A,B,i} C^{(A)}_{B,i}(t,x) \frac{\partial u_{B}(t,x)} {\partial x_{i}} e_{A}
|
(0.2) |
The desired function
u(t,x) = åBuB(t,x)eBu(t,x) = \sum_B{u_{B}(t,x)e_{B}} defined in
[0,T]×W Ì R+0 ×Rn+1[0,T]\times\Omega \subset R^{+}_{0} \times R^{n+1} is a C
lifford-algebra-valued function with real-valued components
uB(
t,
x). We give sufficient conditions on the coefficients of the operator
L{\mathcal{L}} under which
L{\mathcal{L}} is associated to the C
auchy-R
iemann operator
D{\mathcal{D}} of C
lifford analysis. For such an operator
L{\mathcal{L}} the initial value problem (0.1) is solvable for an arbitrary monogenic initial function
u0 and the solution is also monogenic for each
t.
Mathematics Subject Classification (2000). Primary 35F10 - Secondary 30G35
Keywords. Initial value problem - monogenic function - scales of Banach spaces
Received: March 06, 2008. Accepted: July 04, 2008.