Starting from a finitely ramified self-similar set X we can construct an unbounded set X<∞> by blowing-up the initial set X. We consider random blow-ups and prove elementary properties of the spectrum of the natural Laplace operator on X<∞> (and on the associated lattice). We prove that the spectral type of the operator is almost surely deterministic with the blow-up and that the spectrum coinc...

In this paper, we study positive blow-up solutions of the semilinear parabolic system with localized reactions ut = Δu+ vr + up(0,t), vt = Δv + us + vq(0,t) in the ball B = {x ∈ R N : |x| < R} , under the homogeneous Dirichlet boundary condition. It is shown that nonsimultaneous blow-up may occur according to the value of p , q , r , and s ( p,q,r,s > 1). We also investigate blow-up rates of al...

We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the e...

Formation of blow-up singularities for the Navier–Stokes equations (NSEs) ut + (u · ∇)u = −∇p+∆u, divu = 0 in R × R+, with bounded data u0 is discussed. Using natural links with blow-up theory for nonlinear reaction-diffusion PDEs, some possibilities to construct special self-similar and other related solutions that are characterized by blow-up swirl with the angular speed near the blow-up time...

Blow-up phenomena for solutions of some nonlinear parabolic systems with time dependent coefficients are investigated. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.

Let Ω be a bounded smooth domain in R . We consider the problem ut = ∆u + V (x)u in Ω × [0, T ), with Dirichlet boundary conditions u = 0 on ∂Ω × [0, T ) and initial datum u(x, 0) = Mu0(x) where M ≥ 0, u0 is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of M . As a consequence of these estimates we find that, for M larg...

We present adaptive procedures in space and time for the numerical study of positive solutions to the following problem,  ut(x, t) = (u)xx(x, t) (x, t) ∈ (0, 1)× [0, T ), (u)x(0, t) = 0 t ∈ [0, T ), (u)x(1, t) = u(1, t) t ∈ [0, T ), u(x, 0) = u0(x) x ∈ (0, 1), with p,m > 0. We describe how to perform adaptive methods in order to reproduce the exact asymptotic behavior (the blow-up rate an...

We study solutions of some supercritical parabolic equations which blow up in finite time but continue to exist globally in the weak sense. We show that the minimal continuation becomes regular immediately after the blow-up time and if it blows up again, it can only do so finitely many times.

In this article we study a normalized double obstacle problem with polynomial obstacles p ≤ p under the assumption p(x) = p(x) iff x = 0. In dimension n = 2 we give a complete characterization of blow-up solutions depending on the coefficients of the polynomials p, p. We see that there exists a new type of blow-ups, that we call double-cone solutions since the noncoincidence set is a union of t...

We consider operations that change the size of images, either shrinks or blow-ups. Image processing offers numerous possibilities, put at everyone's disposal with such computer programs as Adobe Photoshop. We consider a different class of operations, aimed at immediate visual awareness, rather than pixel arrays. We demonstrate cases of blow-ups that do not sacrifice apparent resolution. This ap...