Higher-Order Semilinear Parabolic Equations

• Higher-order semilinear parabolic equations are partial differential equations that extend the heat equation by incorporating higher-order diffusion operators and nonlinear terms, leading to phenomena like finite-time blow-up.
• They display rich dynamics such as self-similar blow-up profiles, critical Fujita-type exponents, and intricate spectral properties that govern the transition between global existence and blow-up.
• Advanced numerical methods, including high-order compact schemes and implicit multistep techniques, are crucial for accurately capturing the oscillatory and singular behaviors inherent in these equations.

A higher-order semilinear parabolic equation is a partial differential equation of the form

$$\partial_t u = -(-\Delta)^m u + |u|^{p-1}u,\quad x\in\mathbb{R}^N,\; t>0,$$

where $m\geq1$ is an integer (often $m\geq2$ for "higher-order" classification), $\Delta$ is the Laplacian, and $p>1$. These equations generalize the classical semilinear heat equation to higher-order dissipative operators and arise in various contexts including phase transitions, thin film dynamics, and pseudoparabolic flow. They are characterized by rich phenomena such as finite-time blow-up, critical Fujita-type exponents, existence and nonexistence thresholds, oscillatory self-similar solutions, and intricate spectral properties of the associated linearized operators.

1. Problem Formulations and Critical Exponents

A prototypical Cauchy problem for higher-order semilinear parabolic equations is

$$\partial_t u + (-\Delta)^m u = |u|^p, \quad u(x,0)=u_0(x)\geq0,$$

with $u_0$ a nonnegative Radon measure or $L^1\cap L^\infty$ function on $\mathbb{R}^N$ (Ishige et al., 2019, Galaktionov et al., 2012). The sign convention for the diffusion term varies; some works use the generator $A=-(-\Delta)^m$.

The critical Fujita exponent, which governs global existence versus blow-up, is

$p_F = 1+\frac{2m}{N},$

generalizing the classical case $p_F=1+\frac2N$ for $m=1$ (Galaktionov et al., 2012, Ishige et al., 2019). For $1<p<p_F$, all nontrivial nonnegative solutions blow up in finite time for a wide class of initial data; for $p>p_F$, global solutions may exist even for large data.

2. Blow-up Dynamics and Self-similarity

Type I blow-up is characterized by the pointwise bound

$$c(T-t)^{-1/(p-1)} \leq \|u(\cdot,t)\|_\infty \leq C(T-t)^{-1/(p-1)}$$

for some $T<\infty$ (Ghoul et al., 2018). Transforming to similarity variables

$$s = -\ln(T-t), \quad y = x (T-t)^{-1/(2m)}, \quad w(y,s) = (T-t)^{1/(p-1)}u(x,t),$$

the rescaled profile $w(y,s)$ evolves according to

$$\partial_s w = -(-\Delta)^m w - \frac{1}{2m}y\cdot\nabla w - \frac{1}{p-1}w + |w|^{p-1}w,$$

which admits formal profiles $\Phi(z)$ with $z = y s^{-1/(2m)}$ solving

$$-\frac{z}{2m}\cdot\nabla\Phi - \frac{1}{p-1}\Phi + \Phi^p = 0,$$

yielding explicit blow-up profiles of the form

$$\Phi(z) = \kappa\ (1 + B_{m,p} |z|^{2m})^{-1/(p-1)}\ ,\quad \kappa = (p-1)^{-1/(p-1)}.$$

Rigorous construction employs spectral decomposition of a non self-adjoint operator, reduction to a finite number of unstable modes, and a topological degree argument to control the solution in a shrinking set (Ghoul et al., 2018).

3. Spectral Theory and Asymptotic Analysis

The linearized operator in similarity variables,

$$L_m = -(-\Delta)^m - \frac{1}{2m} y\cdot\nabla + \mathrm{Id},$$

has real, simple spectrum

$$\operatorname{Spec}(L_m) = \{1 - |\beta|/(2m)\ :\ \beta\in\mathbb{N}^N\}$$

with eigenfunctions $\psi_{\beta}$ forming a Hermite-like basis in suitable exponentially weighted $L^2$ spaces. The adjoint spectrum is identical, with dual eigenfunctions generated via derivatives of a rescaled fundamental solution (Ghoul et al., 2018, Galaktionov et al., 2012). The sectoriality and positivity of the semigroup $e^{sL_m}$ are established by explicit kernel representations.

This structure underpins the decomposition of deviations from the approximate blow-up profile into unstable (inner) modes, a rapidly decaying infinite-dimensional component, and a spatially localized outer part. Modulation equations for the coefficients of unstable modes yield polynomial or logarithmic decay rates, while spectral gaps enable control of higher modes.

4. Existence Theory and Singular Initial Data

The sign-changing fundamental solution for $m\geq2$ obstructs maximum principle strategies. Existence for general nonnegative Radon measures is established via domination by a positive majorizing kernel $K$ constructed from fractional heat kernels: $K(x,t) = G_\theta(x, t^{2m})$ for $\theta \in (0,2)$, ensuring order preservation, smoothing, and semigroup properties up to constants (Ishige et al., 2019). For a local solution to exist, necessary smallness/regularity conditions on the initial data $\mu$ are quantified by

$$\sup_{x\in\mathbb{R}^N} \mu(B(x,\sigma)) \leq C \sigma^{N/p - 2m/(p-1)}$$

in the subcritical case, with logarithmic corrections at $p=p_F$. For singularities of the form $\mu(x)\asymp|x|^{-\frac{2m}{p-1}}$, existence depends on the amplitude; the threshold is sharp (Ishige et al., 2019).

For Dirac mass initial data, existence is strictly subcritical: $\mu=D\delta_0$ yields a solution if and only if $1<p<p_F$ (Ishige et al., 2019).

5. Qualitative Theory: Global, Blow-up, and Sign-changing Solutions

In the subcritical range $1<p<p_F$, solutions with $\int u_0(x)dx>0$ blow up in finite time; all nontrivial solutions with suitable initial data in this regime exhibit type I blow-up with the universal rate discussed above (Galaktionov et al., 2012). For $p>p_F$, small data may yield global solutions.

Self-similar and sign-changing solutions are central to the classification theory. Global-in-time self-similar solutions take the form

$$u_S(x,t) = t^{-1/(p-1)} f\left(\frac{x}{t^{1/(2m)}}\right),$$

with $f$ solving a semilinear elliptic equation. The spectral bifurcation structure determines branches of such solutions, with countable sequences of critical exponents $p_\ell = 1 + 2m/(n+\ell)$. For even $\ell$, transcritical bifurcations yield self-similar solutions with explicit nodal structure; for odd $\ell$, bifurcations are typically of pitchfork or more exotic type, with global and local continuation established via Lyapunov-Schmidt reductions and numerical continuation (Galaktionov et al., 2012).

Non-self-similar, stable-manifold solutions exhibit polynomial decay and are constructed around linearly stable modes of the linearized equation in similarity variables.

6. Nonlinear Memory Effects and Blow-up Rates

Incorporating a nonlocal-in-time (memory) term yields the equation

$$u_t + (-\Delta)^m u = \int_0^t (t-s)^{-\gamma} |u(x,s)|^p ds, \quad 0<\gamma<1,$$

for which sharp blow-up rates are derived using Liouville-type theorems for entire bounded solutions of the infinite memory equation (Fino, 2020). The blow-up rate is

$$\sup_{x}|u(x,t)| \asymp (T^*-t)^{-\alpha_1}, \quad \alpha_1 = \frac{2-\gamma}{p-1},$$

with matching upper and lower bounds for blow-up, conditional on $p\le 1+2m(2-\gamma)/(n-2m+2m\gamma)$ or $p<1/\gamma$. The approach employs rescaling, profile convergence, and contradiction arguments resting on fractional integration and cut-off techniques.

7. Numerical Methods for Higher-Order Semilinear Parabolic Equations

Efficient numerical approximation of higher-order semilinear parabolic equations leverages high-order compact difference schemes in space and implicit multistep methods (BDF2) in time. Two-grid approaches combine a nonlinear coarse-grid solve with a fine-grid linearized correction, using bi-cubic Lagrange interpolation for mapping between grids (Zhang et al., 2023).

Error bounds are established under admissible mesh and time-step conditions:

• Fourth-order spatial accuracy ($O(h^4)$)
• Second-order temporal accuracy ($O(\tau^2)$), provided $\tau = o(h^{1/2})$
• Optimal accuracy retained for the two-grid scheme when $H = O(h^{4/7})$ and $\tau = o(H^{1/2})$

These results ensure efficient computation of complex blow-up dynamics and self-similar structures, validated by numerical tests demonstrating convergence and computational savings.

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This comprehensive framework encompasses existence, blow-up, qualitative behavior, and computational aspects of higher-order semilinear parabolic equations, illustrating the interplay between spectral theory, nonlocal analysis, and numerical approximation (Ghoul et al., 2018, Galaktionov et al., 2012, Ishige et al., 2019, Fino, 2020, Zhang et al., 2023).