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| WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZEDDIFFUSION EQUATIONS WITH REACTION |
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Citation： |
Wang Junyu.WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZEDDIFFUSION EQUATIONS WITH REACTION[J].Chinese Annals of Mathematics B,1994,15(3):283~292 |
| Page view： 930
Net amount： 671 |
Authors： |
Wang Junyu; |
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| Abstract： |
The author demonstrate that the two-point boundary value problem
$$ \cases
p'(s) = f'(s) - \la p^\be(s) & \text{for } s \in (0,1); \be \in (0,1),
\ p(0) = p(1) = 0, p(s) > 0 & \text{if } s \in (0,1),
\endcases $$
has a solution $(\bla, \bar p(s))$, where $|\bla|$ is the smallest
parameter, under the minimal stringent restrictions on $f(s)$, by
applying the shooting and regularization methods. In a classic paper,
Kolmogorov et. al. studied in 1937 a problem which can be converted into
a special case of the above problem.
The author also use the solution $(\bla, \bar p(s))$ to construct a weak
travelling wave front solution $u(x,t) = y(\xi)$, $\xi = x - Ct$, $C =
\bla N / (N+1)$, of the generalized diffusion equation with reaction
$$ \dpa{}{x} \left( k(u) \left| \dpa u x \right|^{N-1} \dpa u x \right)
- \dpa u t = g(u), $$
where $N>0$, $k(s)>0$ a.e. on $[0,1]$, and $f(s) := \f{N+1}{N} \int_0^s
g(t) k^{1/N}(t) dt$ is absolutely continuous on $[0,1]$, while $y(\xi)$
is increasing and absolutely continuous on $(-\infty, +\infty)$ and
$$ (k(y(\xi)) |y'(\xi)|^N)' = g(y(\xi)) - C y'(\xi) \quad \text{a.e. on
} (-\infty, +\infty), $$
$$ y(-\infty) = 0, \qquad y(+\infty) = 1. $$ |
Keywords： |
Generalized diffusion equation, Weak travelling wave
front solution,Two-point boundary value problem, Shooting method,
Regularization method. |
Classification： |
35K57. |
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