Oscillatory blow-up and gradient estimates for semilinear heat equations

Abstract: For reaction-diffusion with blow-up nonlinearities, we consider the question whether the sup norm of any positive blow-up solution must be eventually monotone nondecreasing in time. While some sufficient conditions are known, especially for radial solutions, this natural and basic question for the blow-up theory does not seem to have been addressed so far in full generality. We construct surprising (nonradial) counter-examples of blow-up solutions with oscillatory $L<sup>\infty$ norm, for any Sobolev supercritical power nonlinearity, which show that this property may fail. In addition, this provides examples of type II blow-up for any supercritical power, which considerably increases the known range of powers for which type II blow-up may occur. Moreover, whereas all the type II blow-up rates known so far were at most polynomial, the blow-up in our counter-examples can be arbitrarily singular. As a related question, we clarify the gradient estimates obtained and used in previous works. In particular we show that these estimates hold only at times when the $L<sup>\infty$ norm is maximal with respect to the past.

• The paper constructs nonradial positive solutions for every Sobolev-supercritical power whose sup norm oscillates between prescribed levels as blow-up occurs at spatial infinity.
• The construction produces type II blow-up exceeding the standard rate by any prescribed factor, while also yielding type I oscillatory examples in part of the sub-Lepin range.
• The paper corrects the Friedman–McLeod gradient estimate by showing it requires the current sup norm to equal the maximum over all past times, and establishes eventual monotonicity below the Sobolev critical exponent.

Overview and main question

This paper by Quittner and Souplet addresses a basic structural question in the blow-up theory of semilinear parabolic equations: for the problem

$$u_t - \Delta u = f(u), \qquad u|_{\partial\Omega}=0, \qquad u(x,0)=u_0(x)\ge 0,$$

with $f\in C^1([0,\infty))$, $f(0)\ge 0$, $f(s)>0$ for $s>0$, must the sup norm $U(t):=\|u(\cdot,t)\|_\infty$ of a positive blow-up solution become nondecreasing close to the blow-up time? A positive answer would seem plausible on heuristic grounds, since blow-up signals that the reaction term eventually dominates diffusion near maximum points. Prior to this work, only sufficient conditions were known (e.g., $\Delta u_0 + f(u_0)\ge 0$, radial nonincreasing data in a ball), and no counterexample existed.

The paper answers the general question negatively. It constructs nonradial blow-up solutions with oscillatory $L^\infty$ norm for every Sobolev supercritical power $p>p_S$ in $\mathbb{R}^n$, $f\in C^1([0,\infty))$0. As corollaries, it produces type II blow-up for every supercritical power — considerably enlarging the previously known range (previously restricted to $f\in C^1([0,\infty))$1 with $f\in C^1([0,\infty))$2, isolated exponents such as $f\in C^1([0,\infty))$3 for $f\in C^1([0,\infty))$4 or $f\in C^1([0,\infty))$5 for $f\in C^1([0,\infty))$6, or sign-changing critical cases) — with rates that are arbitrarily singular, whereas all previously known type II rates were at most polynomial. The paper also clarifies the status of the classical Friedman–McLeod gradient estimate, showing that its validity requires the sup norm to be maximal with respect to the past.

Nonmonotone blow-up: main construction

The central result states that for $f\in C^1([0,\infty))$7, $f\in C^1([0,\infty))$8, $f\in C^1([0,\infty))$9 with $f(0)\ge 0$0, there exist initial data $f(0)\ge 0$1, $f(0)\ge 0$2, whose solution blows up at finite time $f(0)\ge 0$3 with an oscillating sup norm: there are sequences $f(0)\ge 0$4 and times $f(0)\ge 0$5 such that, for any prescribed increasing sequence $f(0)\ge 0$6,

$f(0)\ge 0$7

Thus the oscillations can be made arbitrarily strong, tracking any preassigned sequence of peaks. An important feature is that blow-up occurs at space infinity: the maxima are achieved at points $f(0)\ge 0$8 tending to infinity, which distinguishes these examples from all prior type II constructions, where blow-up occurs at a finite point or on a manifold.

The proof proceeds by superposing translated copies of a "threshold" solution $f(0)\ge 0$9 with incomplete $f(s)>0$0 blow-up. For $f(s)>0$1 (the Lepin exponent), $f(s)>0$2 is the explicit peaking solution built from the profile functions $f(s)>0$3 solving the associated self-similar ODEs; for $f(s)>0$4, $f(s)>0$5 is the threshold solution obtained as the limit of solutions with scaled radial data, known from Matano–Merle to have incomplete blow-up. Each building block $f(s)>0$6 has a sup-norm peak of height $f(s)>0$7 at time $f(s)>0$8, followed by decay below level $f(s)>0$9 before the next peak. The centers $s>0$0 are spaced so that the supports' regions $s>0$1 recede rapidly to infinity. The core technical step is a trapping-region argument: writing $s>0$2 via the Duhamel formula over the partition $s>0$3, the authors bound each contribution using the spatial decay $s>0$4 and Gaussian decay of the heat kernel, controlling the critical adjacent-region terms through a carefully designed piecewise linear barrier $s>0$5. This yields $s>0$6, whence $s>0$7, transferring the oscillation pattern to $s>0$8 and forcing $s>0$9.

Two refinements deserve note. First, for $U(t):=\|u(\cdot,t)\|_\infty$0 the construction can be tuned so that the blow-up is of type I (choosing $U(t):=\|u(\cdot,t)\|_\infty$1, $U(t):=\|u(\cdot,t)\|_\infty$2); moreover, type I oscillatory examples in dimension $U(t):=\|u(\cdot,t)\|_\infty$3 with $U(t):=\|u(\cdot,t)\|_\infty$4 extend trivially to higher dimensions by cylindrical extension. Second, the solution admits a classical continuation past $U(t):=\|u(\cdot,t)\|_\infty$5, though not continuous into $U(t):=\|u(\cdot,t)\|_\infty$6 since $U(t):=\|u(\cdot,t)\|_\infty$7.

Type II blow-up for all supercritical powers

Combining the construction with the universal lower rate estimate $U(t):=\|u(\cdot,t)\|_\infty$8, where $U(t):=\|u(\cdot,t)\|_\infty$9 and $\Delta u_0 + f(u_0)\ge 0$0, and choosing $\Delta u_0 + f(u_0)\ge 0$1 for any increasing $\Delta u_0 + f(u_0)\ge 0$2 with $\Delta u_0 + f(u_0)\ge 0$3, the authors obtain times $\Delta u_0 + f(u_0)\ge 0$4 with

$\Delta u_0 + f(u_0)\ge 0$5

Consequently the blow-up is of type II, and the rate exceeds $\Delta u_0 + f(u_0)\ge 0$6 along a sequence by an arbitrary margin — the blow-up can be "arbitrarily singular," far beyond polynomial rates. This holds for every $\Delta u_0 + f(u_0)\ge 0$7, which substantially extends the parameter range of known type II phenomena for positive solutions. The trade-off is that these examples blow up at space infinity rather than at a point, so they do not subsume the finite-point constructions; whether analogous oscillatory behavior can occur in bounded domains remains open.

Property (P) and bounded domains

A weaker but more robust formulation asks whether $\Delta u_0 + f(u_0)\ge 0$8 becomes nondecreasing whenever it is large enough: property (P) requires that for each $\Delta u_0 + f(u_0)\ge 0$9 there exists $L^\infty$0 such that $L^\infty$1 and $L^\infty$2 imply $L^\infty$3 nondecreasing on $L^\infty$4. The oscillatory theorem shows (P) fails in $L^\infty$5 for both blow-up and global solutions (the latter via the known grow-up solutions with $L^\infty$6, $L^\infty$7). The paper further shows failure in bounded convex domains for global solutions:

For $L^\infty$8, $L^\infty$9, either $p>p_S$0 with $p>p_S$1 convex bounded, or $p>p_S$2 with $p>p_S$3 and radially symmetric data, there exist arbitrarily small scalings $p>p_S$4 of a fixed profile producing global solutions that exceed any prescribed level $p>p_S$5 at some time yet decay to zero as $p>p_S$6. The proof uses the threshold scaling $p>p_S$7 (whose solution blows up when $p>p_S$8, or is global unbounded when $p>p_S$9) together with continuous dependence: solutions just below threshold pass near the large values of the threshold solution before decaying.

By contrast, the assumption $\mathbb{R}^n$0 is necessary: for $\mathbb{R}^n$1 with $\mathbb{R}^n$2, property (P) holds on any smooth domain, and in fact a stronger quantitative version holds: if $\mathbb{R}^n$3 and $\mathbb{R}^n$4, then for any $\mathbb{R}^n$5 there is $\mathbb{R}^n$6 such that $\mathbb{R}^n$7 and $\mathbb{R}^n$8 imply

$\mathbb{R}^n$9

so that $f\in C^1([0,\infty))$00 follows. The proof is by contradiction and rescaling: violating sequences, after parabolic rescaling about near-maximum points, converge to ancient solutions of $f\in C^1([0,\infty))$01 (in $f\in C^1([0,\infty))$02 or a half-space, the latter symmetrized by odd reflection), which the Liouville-type results of Merle–Zaag force to depend only on time — contradicting the differential inequality. A companion remark shows the pointwise analogue $f\in C^1([0,\infty))$03 near large maxima, an ODE-like behavior consistent with the classification of Merle–Zaag.

Gradient estimates and the correction of Friedman–McLeod

The second theme concerns the estimate underlying several classical applications. With $f\in C^1([0,\infty))$04, $f\in C^1([0,\infty))$05, and $f\in C^1([0,\infty))$06 functions vanishing on $f\in C^1([0,\infty))$07, the correct statement (traceable to Sperb, and implicit in later accounts) is:

If $f\in C^1([0,\infty))$08 is convex, $f\in C^1([0,\infty))$09, and $f\in C^1([0,\infty))$10 in $f\in C^1([0,\infty))$11, then the same inequality holds throughout $f\in C^1([0,\infty))$12. The proof is a maximum-principle argument on the auxiliary quantity $f\in C^1([0,\infty))$13, using the vector field $f\in C^1([0,\infty))$14 and, to handle unbounded domains, subtracting a barrier $f\in C^1([0,\infty))$15 and invoking the Hopf lemma at boundary contact points, where convexity gives $f\in C^1([0,\infty))$16.

Friedman and McLeod claimed this with $f\in C^1([0,\infty))$17 in place of $f\in C^1([0,\infty))$18; the paper shows this claim is false in general. Proposition 3 establishes that the conclusion with $f\in C^1([0,\infty))$19 holds if and only if $f\in C^1([0,\infty))$20: necessity follows because the estimate at any earlier time forces $f\in C^1([0,\infty))$21, while Theorem 1 supplies situations where assumption and conclusion decouple. The source of the original gap is identified precisely: the proof required $f\in C^1([0,\infty))$22 wherever $f\in C^1([0,\infty))$23, which is equivalent to $f\in C^1([0,\infty))$24; the erroneous step was the assertion that $f\in C^1([0,\infty))$25 is monotone increasing by the maximum principle.

In one dimension or radial settings, Proposition 4 sharpens this to a necessary condition at individual times: $f\in C^1([0,\infty))$26 has left and right derivatives everywhere (with $f\in C^1([0,\infty))$27, differentiability failing only countably often, via the zero-set analysis of $f\in C^1([0,\infty))$28), and validity of the estimate $f\in C^1([0,\infty))$29 at time $f\in C^1([0,\infty))$30 implies $f\in C^1([0,\infty))$31. Consequently, without eventual monotonicity of $f\in C^1([0,\infty))$32, the estimate is guaranteed only along a sequence of times $f\in C^1([0,\infty))$33 with $f\in C^1([0,\infty))$34.

The paper audits the classical applications accordingly. In Friedman–McLeod's supercritical-norm blow-up result ($f\in C^1([0,\infty))$35), the argument survives only with $f\in C^1([0,\infty))$36 in place of $f\in C^1([0,\infty))$37, though the full statement is recovered by Weissler's semigroup methods even in nonconvex domains; the related lower bound $f\in C^1([0,\infty))$38 can be salvaged with a degraded constant only under $f\in C^1([0,\infty))$39, an inequality the new counterexamples refute in every left neighborhood of $f\in C^1([0,\infty))$40. Fila–Pulkkinen's type I estimate for exponential growth remains valid because their hypotheses fall under the radial monotonicity case. Lacey's global/regional blow-up result for $f\in C^1([0,\infty))$41 used the estimate at all times near $f\in C^1([0,\infty))$42 and does not appear repairable by these means, although the conclusion itself was recently re-established by entirely different arguments avoiding the estimate.

Limitations and open questions

Several restrictions qualify the results. The counterexamples require $f\in C^1([0,\infty))$43 and rely on blow-up at space infinity; whether strongly nonmonotone blow-up or grow-up of the type constructed here can occur in bounded domains is explicitly left open. Property (P) is established only for powers $f\in C^1([0,\infty))$44 (or asymptotically power-like nonlinearities); for $f\in C^1([0,\infty))$45 with general superlinear $f\in C^1([0,\infty))$46, the failure of (P) remains open, as do the weaker version (Pw) and boundedness of global solutions in nonconvex domains. The gradient estimate of Proposition 2 requires convexity of $f\in C^1([0,\infty))$47 and $f\in C^1([0,\infty))$48, and its applicability near blow-up hinges on monotonicity properties that fail in general. Finally, the type II examples, while valid for all $f\in C^1([0,\infty))$49, do not address whether oscillatory or arbitrarily fast type II blow-up can occur at finite points.

Conclusion

The paper settles a long-standing basic question negatively: the sup norm of a blow-up solution need not be eventually monotone, and may oscillate with arbitrarily prescribed amplitude for every Sobolev supercritical power. The same construction yields type II blow-up across the full supercritical range, with rates exceeding the type I scale by an arbitrary function, albeit through blow-up at space infinity. Complementing these negative results, the paper proves a definitive form of the gradient estimate — valid exactly when the current sup norm dominates its past supremum — thereby correcting the classical Friedman–McLeod statement and delineating precisely which of its historical applications remain sound.