The concept of blow-up is now formulated in its simplest form in the following frame. First analysis of the most striking effect of spatial localization of upwelling boundary regime (S-regime of upwelling generated by an upwelling standing wave) in quasilinear diusion equations was carried out by Samarskii and Sobol' in 1963 152]. The occurrence and type of blow-up depends on the parameter >0, initial data and domain.
We then concentrate on the analysis of the main questions raised in the study of explosions for reaction-dilution equations. Self-inflation then becomes the usual form of inflation and xes the inflation rates. This is the problem of continuation after detonation, also known as beyond detonation.
To point out the interest in the study of continuation after inflation is one of the main concerns of these notes. Countable spectra of inflation patterns for ut= (juxjux)x+eu were obtained in 31] where transition phenomena between non-linear and semi-linear (see below) spectra of inflation patterns were described. Let us explain the main lines of the region inflation result as described in 85].
Then the inflation occurs in the night time and the inflation group is exactly the interval.
Continuation after blow-up
When the solution inflates at time T >0, we can think of the inflation set in the usual way, cf. The set is interesting when inflation is incomplete, since it will be a subset of , generally nonvoid for t > T. Under suitable generic assumptions on , f and the initial data u0 we obtain a rather complete characterization of the non-trivial continuation of solutions of the Cauchy problem after inflation in terms of the properties of the constitutive nonlinearities and onuf0).
A natural problem is establishing the conditions for complete/incomplete inflation in terms of component functions. Precisely, inflation for positive constant initial data occurs if and only if the integral I1(u) =Z u. The blow-up and continuation problem is thus completely solved and the result does not depend on the data.
The presence of complete or incomplete blooming depends on the behavior of the three integrals at infinity. For non-initial data, the necessity of this condition for increase comes from the maximum principle. The proofs given in 87] extend to solutions with rather general initial data, but we need some additional conditions on the scaling approach in the statement of the results.
In fact, the analysis of complete/incomplete blasting is local in the sense that the behavior fort >. T depends only on the behavior of the solution in a small neighborhood to a given bursting point, and is thus independent of the boundary conditions. In the incomplete burst, the singular propagation speed T is assumed to be nit.
In a first step we perform a careful analysis of the existence and properties of the traveling wave (TW) family of solutions, and a subsequent step allows us to validate the inflation results using those waves. 3a) Classification of traveling waves. Finally, incomplete inflation leads to a two-phase problem: while in the spacetime region Q1 = Bu](t) IR+ the proper solution u(xt) is infinite, in the complement Q2 = fIRnBu](t)g IR+ the solution is nite and the equation holds. Furthermore, there exists a class of initial data such that for inflation interfaces s(t) the optimal regularity for t > T(u0) is C11 (s0(t) is Lipschitz continuous) but not C2.
On transient blow-up patterns in several dimensions
The construction relies on the study of the stability of the ODE for the singular stationary solution S, which is completely different for p < pu and ppu. We claim that, at least for self-similar solutions, the upper value of pp is optimal for the theorem, see the example below. They are also weak solutions of the equation in the standard sense of integration by parts.
In connection with this result, let us mention that it is explained in 137] that for ppsequation (1.7) posed in a bounded, smooth, convex domain with Dirichlet boundary condition and a special choice of non-negative initial data admits a global L1 solution ~u(xt) which is not bounded in L1(). It follows from the result of full blast in the critical case Sobolev 91], Sect. We prove that in the range p 2(pspp) such symmetric solutions explode in no time.
We first construct self-similar inflation solutions for t < T as a first step in proving the existence of global peak solutions. The existence of a nontrivial solution > 0 in the subcritical range 1 < p < ps has been established in 151], Chapter 4, p= ps was studied in Sect. The evidence is based on a detailed ODE analysis of the intersection properties of the solutions of (7.3) with respect to the single stationary prole S, 91], Sect.
We now show that the self-similar solutions constructed above can be continued in a non-trivial way as real solutions beyond T. The blown self-similar solutions constructed in the previous two theorems have a non-trivial self-continuation for t > T and indeed u(rt)S(x) in IRn (T1): They are also self-similar for t > T. Then the proper continuation for t > T of the blown self-similar solution coincides with the comparable solution (7.6) with a specially chosen prole f().
Note that the correct solution with singular stationary data S does not decrease with time in this case. Then the proper continuation of the self-similar solution of the blow-up (7.3) with prole1() satisfying (7.5) is self-similar and bounded for t > T.
Extinction
If p > ;m, such maximal (weak) continuity exists and the Cauchy problem with compactly supported initial data makes sense. The asymptotic behavior of such extinction extinction behavior was proved to be approximately similar 86] in the critical case = 2; m. The interface propagation in this case is studied in 83], where analyticity is proved and a countable spectrum of curve and injection interface models is constructed with the local spatial form governed by Kummer polynomials.
It was found that the singular interface propagation is the most interesting in the subcritical region p 2 (;m2 ;m), where the interface equation was found to be of second order 82], in contrast to the first-order Darcy's law for PME. In the semilinear case m = 1, p < 1, the asymptotic extinction behavior is completely different and corresponds to that of inflation for the semilinear heat equation (1.7) 104]. For quasi-linear diusion operators such as (um)xx, such behavior can be self-similar (non-linear patterns) as in the blow-up for quasi-linear heat equations.
In the Cauchy problem in IRn(0T) for m2(0(n;2)=n) there exists a self-similar solution of the second kind and it is asymptotically stable for symmetric solutions 80].
On singularities in mean curvature and curve shortening ows
Finite-time extinction due to the singularity of the equation and the boundary conditions (without the zero-order absorption term) was first demonstrated in 149] (n = 1) for the fast diffusion equation ut = um, m 2 (01), in a bounded domain with u = 0 on. For non-convex curves, the singularity structure is more complex, the peak singularities increase rapidly at a rate of 6]. Localization of blow-up solutions with Dirichlet boundary conditions u(at) = 0, a > =2 was first demonstrated in 56], where the localization length was shown to be 0 for rather arbitrary initial data u0 .
The shortening of the curve is done by the equation V = k+f(y) (f(y) is a given function on the fxyg plane, V denotes the upward velocity C(t) =fy =u(xt)g) reduces to a quasi-linear equation.
On blow-up due to nonlinear boundary conditions
Once appropriate estimates ofu(xt) on @ are available, the results on localization and inflation patterns follow. The existence of localized swelling functions is proven for a general heat conduction equation ut = (k(u)ux)x k(u) > 0 for u > 0 regardless of the behavior of k(u) as u. We refer to the survey paper 129] and the analysis of critical Fujita exponents for different quasilinear equations in 1D 78], see also a partial survey of recent results in 79].
Fully nonlinear and nonlocal parabolic equations
On blow-up in free boundary problems
On blow-up in parabolic systems
In such systems magnification in variables su and v can be different, say single point in u and global in v, see p. Self-similar inflation patterns (unstable) exist for a system with reaction absorption terms. the concentration distribution is negligible in the second equation), are available in Chapter 7 in 151], where a list of related references is available.
On blow-up in higher-order parabolic equations
Blow-up in other nonlinear PDEs
Magnification for the generalized Korteweg-de Vries equation ut+upux+uxxx= 0 (p is natural). and for the fifth-order equation with derivative uxxxxx) were studied in 25]. Galaktionov, Stability and blow-up spectra in quasilinear gradient diffusivity problems, Proc. Matano, Convergence, asymptotic periodicity and endpoint blow-up in one-dimensional semilinear heat equations, J.
Galaktionov, On new exact swelling solutions for nonlinear source heat conduction equations and applications, Dier. Vazquez, Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Vazquez, Necessary and sufficient conditions of full blow-up and extinction for one-dimensional quasilinear heat equations, Arch.
Vazquez, Blow-up for quasilinear heat equations described by nonlinear Hamilton-Jacobi equations, J. Vazquez, Blow-up of a class of solutions with free bounds for the Navie-Stokes equations, Adv. Merle, Concentration properties of inflated solutions and instability results for Zakharov equations in dimension two.
Sulem and P.L. Sulem, Blow-up velocity for solutions of the nonlinear Scrodinger equation at critical dimension, Phys. Rossi, Blow-up and Fujita-type curves for a degenerate parabolic system with nonlinear boundary conditions, Preprint, 1999. Mikhailov \Blow-up and Quasilinear Parabolic Equations", Nauka, Moscow, 1987 English translation: Walter de Gruyter, Berlin/New York, 1995.
Suppe, Uniform blow-up proles and boundary behavior for diusion equations with non-local nonlinear source, J. Tzanetis, Asymptotic behavior and blow-up of some unbound solutions to a semilinear heat equation, Proc. Velazquez, Estimates of the (N;1)-dimensional Hausdor measure of the blow-up set for a semilinear heat equation, Indiana Univ.
Herrero, Spatial structure near the blow-up point for semilinear heat equations: a formal approach, USSR Comput. Xin, Augmentation of smooth solutions of the incompressible Navier-Stokes equation with a compact density, Comm.
