Boundedness and infinite-time blow-up of global solutions of the superlinear heat equation at and above the critical Sobolev exponent

For the semilinear heat equation $\partial_t u-\Delta u+mu=g(u)$ on a bounded domain with Dirichlet boundary conditions and a superlinear nonlinearity $g$ of power growth $p$, Cazenave and Lions (1984) proved that global solutions are uniformly bounded below the critical Sobolev exponent $p_S=(N+2)/(N-2)$ and recorded that for $p\ge p_S$ there exist global solutions bounded in $L^2$ and averaged gradient norms yet unbounded in $L^\infty$, leaving open whether this unboundedness develops at a finite time or only as $t\to\infty$, and whether their boundedness theorem extends to the critical exponent itself. Later literature resolved the source questions in the main regimes: unbounded global solutions exist at and above $p_S$; at the critical exponent they are classical solutions whose $\|u(\cdot,t)\|_\infty$ diverges as $t\to\infty$ (grow-up, also called infinite-time blow-up), with rates known in radial and several non-radial settings; and quantitative a priori bounds uniform over bounded sets of initial data are known for power-type nonlinearities throughout the subcritical range. Precise open cores remain: positive non-radial grow-up solutions for the critical Dirichlet problem in dimension four; quantitative a priori bounds for nonlinearities satisfying only the general Cazenave–Lions growth and superlinearity conditions in the exponent range between $p_{CL}=1+12/(3N-4)$ and $p_S$; and the boundedness-versus-grow-up classification for supercritical exponents on general domains without symmetry assumptions.

Background

Consider the initial-boundary value problem for the semilinear heat equation

$$\partial_t u-\Delta u+mu=g(u)\quad\text{in }\Omega\times(0,\infty),\qquad u=0\quad\text{on }\partial\Omega\times(0,\infty),$$

where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $m\ge0$ (more generally $m>-\lambda_1$, with $\lambda_1$ the first Dirichlet eigenvalue of $-\Delta$, so that the linear part is coercive), and $g\in C^1(\mathbb{R})$ is superlinear, meaning $xg(x)\ge(2+\varepsilon)G(x)$ for some $\varepsilon>0$ where $G(x)=\int_0^x g(s)\,ds$, and has power-type growth $|g(x)|\le A|x|+B|x|^p$. A solution blows up in finite time if $\|u(\cdot,t)\|_{L^\infty(\Omega)}$ becomes infinite at a finite time; a solution is global if it exists as a classical solution for all $t>0$. Two quantitative questions organize the theory: whether global solutions are uniformly bounded, $\sup_{t\ge0}\|u(\cdot,t)\|_\infty<\infty$, and whether the bound is quantitative, i.e. $\sup_{t\ge0}\|u(\cdot,t;u_0)\|_\infty\le C$ with $C$ depending only on a norm of the initial datum $u_0$, uniformly over bounded sets of data. The threshold for both is the critical Sobolev exponent $p_S=(N+2)/(N-2)$ for $N\ge3$ (with $p_S=\infty$ for $N\le2$), the exponent at which the Sobolev embedding $H_0^1(\Omega)\hookrightarrow L^{p+1}(\Omega)$ loses compactness.

Cazenave and Lions (1984), improving an earlier result of Ni, Sacks and Tavantzis (1984), proved by energy and interpolation arguments that under the above superlinearity and growth conditions every global solution is uniformly bounded in the $C^2(\overline{\Omega})$ norm for times bounded away from the initial time: quantitatively (the constant depending only on a norm of the initial datum) for $p<p_{CL}:=1+12/(3N-4)$, and with unspecified dependence on the initial datum for $p<p_S$; they also described how the initial-data space splits into a region of bounded global solutions converging to steady states and a region of finite-time blow-up, this decomposition being restricted by the growth condition to $p<p_{CL}$. In the opposite regime $p\ge p_S$ they recorded, adapting a construction of Ni, Sacks and Tavantzis, that on a star-shaped domain there is a nonnegative global solution that is bounded in $L^2$, satisfies averaged gradient bounds and has square-integrable time derivative, yet is unbounded in $L^\infty$ and in every $H^{1+\delta}$; Ni, Sacks and Tavantzis had shown that on convex domains the quantitative estimate fails for $p\ge p_S$. Cazenave and Lions left open whether the unboundedness of the stronger norms of such solutions occurs at a finite time or only as $t\to\infty$, and whether their boundedness theorem extends to the critical exponent $p=p_S$ itself.

The subsequent theory introduced the following notions. A global classical solution with $\|u(\cdot,t)\|_\infty\to\infty$ as $t\to\infty$ is said to blow up in infinite time, or to grow up. For the model power nonlinearity at $p=p_S$ (the energy-critical heat equation, for instance $\partial_t u=\Delta u+u^5$ in $\mathbb{R}^3$), the stationary equation on the whole space, $-\Delta U=U^{(N+2)/(N-2)}$, is solved exactly by the Aubin–Talenti bubbles $U_{\mu,\xi}(x)=\big(N(N-2)\big)^{(N-2)/4}\left(\mu/(\mu^2+|x-\xi|^2)\right)^{(N-2)/2}$ with scale $\mu>0$ and center $\xi\in\mathbb{R}^N$, and unbounded global solutions on bounded domains have the asymptotic form of a concentrating bubble ($\mu\to0$) centered at a spike point (bubbling). The admissible spike locations are constrained by the Robin function $H_\gamma(q,q)$, the diagonal of the regular part of the Green function of $-\Delta-\gamma$ for $\gamma\in(0,\lambda_1)$: in dimension three, for instance, at points $q$ satisfying $3\gamma(q)<\lambda_1$, where $\gamma(q)=\sup\{\gamma>0:H_\gamma(q,q)>0\}$. Blow-up is of type I if $(T-t)^{1/(p-1)}\|u(\cdot,t)\|_\infty$ stays bounded as the blow-up time $T$ is approached, and of type II otherwise. The threshold solution separates initial data whose solutions decay as $t\to\infty$ from data whose solutions blow up in finite time.

Problem Statement

For the initial-boundary value problem $\partial_t u-\Delta u+mu=g(u)$ on a bounded smooth domain $\Omega\subset\mathbb{R}^N$, $N\ge3$, with Dirichlet boundary conditions, with $m\ge0$ (more generally $m>-\lambda_1$, $\lambda_1$ the first Dirichlet eigenvalue of $-\Delta$) and $g\in C^1(\mathbb{R})$ superlinear ($xg(x)\ge(2+\varepsilon)G(x)$ for some $\varepsilon>0$, where $G(x)=\int_0^x g(s)\,ds$) of power-type growth $|g(x)|\le A|x|+B|x|^p$, determine the structure of the set of global solutions (classical solutions existing for all $t>0$) across the critical Sobolev threshold $p_S=(N+2)/(N-2)$. Concretely: (1) for which combinations of exponent $p$, dimension $N$ and domain geometry are all global solutions uniformly bounded, and for which do unbounded global solutions exist, in particular at the critical exponent $p=p_S$ and in the supercritical range $p>p_S$; correspondingly, describe how the space of initial data decomposes into data yielding bounded global solutions and data yielding finite-time blow-up, beyond the restricted exponent range in which this decomposition is currently established; (2) for the unbounded global solutions, determine the divergence of the strong norms ($L^\infty$ and $H^{1+\delta}$) as $t\to\infty$ (the finite-time alternative being empty, since a global solution is by definition a classical solution existing for all $t>0$) and determine their asymptotic profiles and growth rates; (3) establish quantitative a priori bounds $\sup_{t\ge0}\|u(\cdot,t;u_0)\|_{L^\infty(\Omega)}\le C(\|u_0\|)$, with $C$ depending only on a norm of the initial datum, uniformly over bounded sets of initial data, for general nonlinearities $g$ satisfying only the above superlinearity and growth conditions, in the exponent range $p_{CL}\le p<p_S$, where $p_{CL}=1+12/(3N-4)$ and where the known bootstrap arguments for power-type nonlinearities do not apply. As of 2026 the following precise parts are open: (i) existence of a positive non-radial grow-up solution of the critical Dirichlet problem in dimension $N=4$ on a bounded domain; (ii) the quantitative a priori bound in (3) for general $g$ in $p_{CL}\le p<p_S$; (iii) the boundedness-versus-grow-up classification for supercritical $p>p_S$ on general bounded domains without symmetry assumptions. These three open parts are precisely the open sub-cases of (1)–(3): (i) is the unresolved existence sub-case of (1) at the critical exponent $p=p_S$ in dimension $N=4$ (existence alone, not the profile-and-rate demand of (2)); (ii) is the whole of (3); and (iii) is the unresolved supercritical case of (1), including its initial-data decomposition and the corresponding part of (2). The already-settled regimes (the subcritical classification, the initial-data decomposition in its established exponent range, and the radial and several non-radial grow-up results) are context.

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