New diffusion models in image processing
Computers & Mathematics with Applications
| Hi-index | 7.29 |
We study the phenomenon of finite time blow-up in solutions of the homogeneous Dirichlet problem for the parabolic equation u"t=div(a(x,t)|@?u|^p^(^x^)^-^2@?u)+b(x,t)|u|^@s^(^x^,^t^)^-^2u with variable exponents of nonlinearity p(x),@s(x,t)@?(1,~). Two different cases are studied. In the case of semilinear equation with p(x)=2, a(x,t)=1, b(x,t)=b^-0 we show that the finite time blow-up happens if the initial function is sufficiently large and either min"@W@s(x,t)=@s^-(t)2 for all t0, or @s^-(t)=2, @s^-(t)@?2 as t-~ and @!"1^~e^s^(^2^-^@s^^^-^(^s^)^)ds=b^-0, a"t(x,t)@?0, b"t(x,t)=0, min@s(x)2 and maxp(x)@?min@s(x).