Inhomogeneous Singular Parabolic Double Phase Equations

• Inhomogeneous singular parabolic double phase equations are nonlinear PDEs with dual-phase structure, exhibiting p- and q-growth modulated by variable spatial-temporal coefficients.
• They employ intrinsic scaling, Caccioppoli inequalities, and caloric approximation to overcome challenges in establishing higher integrability and partial regularity.
• Key results include optimal gap bounds and balance conditions that ensure existence, uniqueness, and regularity of weak and viscosity solutions.

Inhomogeneous singular parabolic double phase equations are nonlinear parabolic partial differential equations (PDEs) that exhibit competing power-type growths modulated by spatially and temporally varying coefficients. These equations model media with variable structure, capturing singular and degenerate (double phase) behavior. The distinctive feature is the presence of two energy-densities—one with $p$-growth, the other with $q$-growth—interpolated via a nonnegative, Hölder or Lipschitz continuous coefficient. The singular regime refers specifically to the case where $p<2$, inducing challenges in regularity and integrability of solutions, especially in the presence of nontrivial source terms (inhomogeneity).

1. Mathematical Structure and Main Models

The canonical inhomogeneous singular parabolic double phase equation takes the form: $$u_t - \operatorname{div}\left(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du\right) = -\operatorname{div}\left(|F|^{p-2}F + a(x,t)|F|^{q-2}F\right)$$ on a parabolic cylinder $$\Omega_T = \Omega \times (0,T) \subset \mathbb{R}^n \times \mathbb{R}$$, with exponents $2n/(n+2) < p \leq 2 < q$ (singular range), and the modulating function $a \in C^{\alpha, \alpha/2}(\Omega_T)$ for $\alpha \in (0,1]$.

The associated density function is $H(z,s)=s^p + a(z)s^q$. In regions where $a(z)\approx 0$, $p$-growth dominates, while in regions with $a(z)>0$, $q$-growth prevails. The inhomogeneity $F$ must satisfy integrability with respect to $H(z,|F|)$. The model may admit variable exponents and more than two phases, but most analytic progress focuses on classical two-phase structure, including the singular case $p<2$ and $q>p$ (Kim et al., 2023, Kim et al., 4 Jan 2026).

2. Notions of Solution and Functional Framework

A function $u$ is a weak solution if:

• $$u\in C(0,T;L^2(\Omega))\cap L^1(0,T;W^{1,1}(\Omega))$$,
• $\int_{\Omega_T} H(z,|Du|)<\infty$, and
• For every $\varphi\in C^\infty_c(\Omega_T)$:

$$\int_{\Omega_T}\left(-u\,\varphi_t + A(z,Du)\cdot D\varphi\right)\,dz = \int_{\Omega_T} B(z,F)\cdot D\varphi\,dz,$$

where $A(z,\xi) = |\xi|^{p-2}\xi + a(z)|\xi|^{q-2}\xi$, $B(z,F)$ similarly.

A viscosity solution (for $1 < p < 2$ or degenerate $p, q$) is a bounded, upper-semicontinuous function satisfying the appropriate test function inequalities at every non-Lebesgue/null set point (Sen et al., 22 Aug 2025). The equivalence between weak and viscosity solutions holds under boundedness, $a(x,t)$ regularity, time-monotonicity of $a$, and further mild assumptions on $a$’s spatial regularity and level sets.

Admissible function spaces include Orlicz-Sobolev spaces $W^H_\mathcal{P}(Q)$ defined via the density $H(x,t,\xi)$, as well as variable exponent Lebesgue and Sobolev spaces if the exponents $p(z), q(z)$ vary (Arora et al., 2021).

3. Higher Integrability and Gap Bounds

The key analytic challenge in the singular regime ($p<2$) is to establish higher integrability for the gradient of weak solutions, given the competing growths. The main theorem asserts: assuming $q \leq p + \alpha(p(n+2)-2n)/(n+2)$ and $a\in C^{\alpha,\alpha/2}$, there exists $\varepsilon_0>0$ such that for every parabolic cylinder $Q_{2r}\Subset Q_T$ and $\varepsilon\in(0,\varepsilon_0)$,

$$\int_{Q_r} H(z,|Du|)^{1+\varepsilon}\,dz \leq c\left[\left(\int_{Q_{2r}} H(z,|Du|)\,dz\right)^{1+\varepsilon} + \int_{Q_{2r}} (1+H(z,|F|))^{1+\varepsilon}\,dz\right]$$

(Kim et al., 2023, Kim et al., 4 Jan 2026).

The admissible gap $q-p$ is bounded above by the Hölder continuity exponent and the scaling deficit: $q - p \leq \frac{\alpha(p(n+2)-2n)}{n+2}$ ("singular scaling deficit"). For weaker time integrability on $u$ (i.e., $u \in C(0,T;L^s(\Omega)),\ s\ge 2$), the optimal threshold interpolates: $$q \leq p + \frac{\alpha\mu_s}{n+s}, \quad \mu_s = \frac{(p(n+2)-2n)s}{4}$$ reproducing known endpoint results in $L^\infty$ or $L^2$ (Kim et al., 4 Jan 2026).

4. Regularity Theory: Lipschitz, Hölder, and Partial Regularity

Spatial Lipschitz regularity is obtained for continuous solutions when $1 < p \leq q \leq p+1$, $a$ bounded, spatially Lipschitz and temporally continuous, and $f$ satisfies continuous growth conditions: $$|u(x,t)-u(y,s)| \leq C\left(|x-y| + |t-s|^\alpha\right)$$ with $\alpha = p/(p+q)$ for $1$\alpha=1/2$ for $p\geq2$ (Sen et al., 22 Aug 2025).

Hölder continuity of the gradient $Du$ is established outside a parabolic measure-zero singular set, provided $q<\min\{p+\alpha p/(n+2), p+1\}$, $a\in C^{0,\alpha,\alpha/2}$ (Ok et al., 4 Oct 2025). The regular set is characterized via Campanato-type excess decay arguments and caloric approximation, with uniform bounds for the "good" set.

5. Intrinsic Scaling, Cylinder Geometry, and Proof Strategies

Analysis exploits intrinsic geometry—cylinders adapted to local energy density. For singular $p<2$, two types are essential:

• $p$-intrinsic cylinder: scaling as $$B_{\lambda^{-1/p}\rho} \times (t_0-\rho^2, t_0+\rho^2)$$, relevant when $a$ is small.
• $(p,q)$-intrinsic cylinder: $$B_\rho\times(t_0-\lambda^{2-p}\rho^2, t_0+\lambda^{2-p}\rho^2)$$, corresponding to strong $a$ (Kim et al., 2023).

In both, $\lambda$ balances the local energy: $\iint H(z,|Du|)+H(z,|F|) \sim \lambda \rho^{n+2}$. Proofs revolve around:

• Caccioppoli inequalities in shifted Orlicz spaces;
• Localized Sobolev/Gagliardo–Nirenberg and Poincaré inequalities;
• Reverse Hölder inequalities iterated via covering lemmas (Vitali/stop-time analysis);
• Approximation of solutions via linearization and excess decay for partial regularity (Kim et al., 2023, Ok et al., 4 Oct 2025, Kim et al., 4 Jan 2026).

6. Existence, Uniqueness, and Second-Order Regularity

Global existence and uniqueness hold for strong solutions with $u_t\in L^2(Q_T)$, provided source $f\in L^2(0,T;W^{1,2}_0(\Omega))$, and the initial data $u_0$ in $W^{1,\mathcal{H}_0(\Omega)}$ (Arora et al., 2021). Global higher integrability of $|\nabla u|^{\min\{p(z),q(z)\}+r}$ for any $r\in (0,4/(N+2))$ is obtained. Second-order differentiability of the flux,

$$D_{x_j} \left( (a|\nabla u|^{p-2} + b|\nabla u|^{q-2})^{1/2} D_{x_i}u \right) \in L^{2}(Q_T),$$

is also proved. The "balance condition" $|p(z)-q(z)|<2/(N+2)$ ensures uniform control of phase oscillation necessary for these results.

7. Open Problems and Optimality

The singular regime ($p\to2n/(n+2)^+$, scaling deficit vanishes) precludes nontrivial double phase behavior—no genuine double phase exists as $q\to p$ is enforced. Extension to fully anisotropic/multi-phase equations, removal of technical hypotheses (e.g., time-monotonicity of $a$), and Calderón-Zygmund theory in the singular parabolic double phase context are open research directions (Sen et al., 22 Aug 2025). Counterexamples confirm the optimality of the stated gap conditions; if $q-p$ exceeds the scaling threshold, higher integrability and regularity can fail (Kim et al., 2023, Kim et al., 4 Jan 2026).

8. Summary Table: Structural and Regularity Results

Reference	Equation Structure	Regularity/Integrability
(Kim et al., 2023)	$$u_t - \operatorname{div}(\dots) = -\operatorname{div}(\dots)$$	Higher integrability of $|Du|$: Reverse Hölder in intrinsic cylinders
(Ok et al., 4 Oct 2025)	Homogeneous system, $H(z,s) = s^p + a(z)s^q$	Partial $C^{1,\alpha}$ regularity outside singular set of measure zero
(Kim et al., 4 Jan 2026)	Inhomogeneous, $p<2<q$	Interpolated gap conditions, global higher integrability
(Arora et al., 2021)	Variable exponents and coefficients	Existence, uniqueness, higher global and second-order regularity
(Sen et al., 22 Aug 2025)	Nonlinear source $f(z,Du)$	Local Lipschitz (space), sharp Hölder (time) regularity

A plausible implication is that the methods developed for the singular parabolic double phase equations—intrinsic scaling, shifted Orlicz inequalities, and caloric approximation—provide a robust analytic framework for degenerate/singular PDEs beyond the two-phase models. However, technical barriers persist in extending full Calderón-Zygmund theory and relaxing balance- and gap-type structural conditions.