Uniqueness and stability of regional blow-up in a porous-medium equation
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2002
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Abstract
We study the blow-up phenomenon for the porous-medium equation in R-N, N greater than or equal to 1,
u(t) = Deltau(m) + u(m),
m > 1, for nonnegative, compactly supported initial data. A solution u(x, t) to this problem blows-up at a finite time (T) over bar > 0. Our main result asserts that there is a finite number of points x(1), ..., x(k) is an element of R-N, with \x(i) - x(j)\ greater than or equal to, 2R* for i not equal j, such that
lim (t-->(T) over bar)((T) over bar - t)(1/m-1)u(t, x) = Sigma(j=1)(k) w(*)(\x - x(j)\).
Here w(*)(\x\) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation Deltaw(m) + w(m) - 1/m-1w = 0 in R-N and R* is the radius of its support. Moreover u(x, t) remains uniformly bounded up to its blow-up time on compact subsets of R-N\boolean ORj=1k (B) over bar (x(j), R*). The question becomes reduced to that of proving that the omega-limit set in the problem v(t) = Deltav(m) + v(m) - 1/m-1v consists of a single point when its initial condition is nonnegative and compactly supported.
u(t) = Deltau(m) + u(m),
m > 1, for nonnegative, compactly supported initial data. A solution u(x, t) to this problem blows-up at a finite time (T) over bar > 0. Our main result asserts that there is a finite number of points x(1), ..., x(k) is an element of R-N, with \x(i) - x(j)\ greater than or equal to, 2R* for i not equal j, such that
lim (t-->(T) over bar)((T) over bar - t)(1/m-1)u(t, x) = Sigma(j=1)(k) w(*)(\x - x(j)\).
Here w(*)(\x\) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation Deltaw(m) + w(m) - 1/m-1w = 0 in R-N and R* is the radius of its support. Moreover u(x, t) remains uniformly bounded up to its blow-up time on compact subsets of R-N\boolean ORj=1k (B) over bar (x(j), R*). The question becomes reduced to that of proving that the omega-limit set in the problem v(t) = Deltav(m) + v(m) - 1/m-1v consists of a single point when its initial condition is nonnegative and compactly supported.