Solvability of a Mixed Problem for a Time-Fractional PDE with Time-Space Degenerating Coefficients

Abstract: In this paper, we investigate the unique solvability of a mixed boundary value problem for a fractional partial differential equation featuring a degenerate coefficient. By introducing a novel operator and applying the method of separation of variables, we establish the existence of eigenvalues and eigenfunctions for the associated spectral problem and prove that the operator possesses a discrete spectrum. Additionally, we establish the relationship between the given data and the unique solvability of the problem, offering new insights into how degeneracy influences fractional diffusion processes.

• The paper demonstrates existence and uniqueness for mixed boundary problems in time-fractional PDEs using spectral decomposition and Caputo-like hyper-Bessel derivatives.
• It highlights how spatial degeneracy impacts boundary conditions and solution regularity, distinguishing between classical and weak solutions based on parameter beta.
• The analysis employs Mittag-Leffler functions and variational methods to provide explicit series solutions, advancing models for anomalous diffusion in heterogeneous media.

Solvability Analysis for Time-Fractional PDEs with Time-Space Degenerating Coefficients

Introduction and Motivation

This paper tackles the unique solvability of mixed-type boundary value problems for a time-fractional partial differential equation (PDE) incorporating time-space degenerating coefficients. The focus is on equations where both the memory effects and spatial heterogeneity are prominent, a context highly relevant to modeling anomalous diffusion, notably in heterogeneous or porous media. Time-fractional derivatives capture nonlocal temporal dynamics, while spatially degenerating coefficients—modeled with powers of the spatial variable—encode variable diffusivity or conductivity, manifesting behavior unattainable using standard PDEs.

The novelty arises from utilizing the Caputo-like counterpart of the hyper-Bessel operator, which generalizes classical fractional differential operators (FDOs) and encompasses the complexity of memory-dependent processes. The main analytical goals are to (i) characterize the spectral properties of the spatial operator with degenerating weights, (ii) resolve the impact of the degeneration degree on admissible boundary conditions, and (iii) rigorously establish existence and uniqueness regimes for classical and weak solutions depending on the degeneracy regime.

Model Formulation

The equation studied has the following structure over $\Omega = \{(x, t) \mid 0 < x < 1,\, a < t \leq T\}$:

$${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$

where $0<\alpha<1$, $\theta<1$, and $0<\beta<2$, $\beta \neq 1$. The initial condition is posed as $u(x, a+)=\varphi(x)$ with inhomogeneous source term $f(x, t)$.

The time-fractional derivative operator is the regularized Caputo-like counterpart of the hyper-Bessel type, parametrized to account for memory effects with power-law kernels and possible time-varying diffusivity. The spatial operator is an elliptic differential operator with degeneration exponent $\beta$, meaning spatial diffusion varies across the domain and can vanish or become singular at the endpoints. The degeneracy introduces subtleties in both function space setting and boundary condition specification.

Boundary Conditions and Degeneration

A key technical step is the divisibility of the analysis based on the value of $\beta$. For $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$0, strong (Dirichlet) conditions must be imposed at $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$1 and $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$2 to ensure well-posedness. In contrast, when $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$3, only a condition at $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$4 is necessary—at $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$5 the natural weight vanishes sufficiently rapidly, obviating the need for an explicit boundary constraint. This dichotomy is made precise via rigorous integration by parts and careful evaluation of boundary terms in the associated variational formulation.

Spectral Theory for the Degenerate Elliptic Operator

The main spectral problem is

$${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$6

with adapted boundary conditions dependent on $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$7. The analysis proves that for all admissible $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$8, the operator is positive definite and symmetric in $${}^C\left((t-a)^\theta \frac{\partial}{\partial t}\right)^\alpha u(x,t)-\frac{\partial}{\partial x}\left(x^{\beta}\frac{\partial u(x,t)}{\partial x}\right)=f(x,t),$$9, and admits a self-adjoint Friedrichs extension. Key functional spaces are weighted Sobolev spaces $0<\alpha<1$0, equipped with nonstandard norms reflecting the degeneracy.

Using variational methods and compact embedding theorems (notably Kondrashov's for weighted classes), the operator is shown to possess a purely discrete spectrum (i.e., a complete orthonormal basis of eigenfunctions and a diverging sequence of eigenvalues), a structural property critical for series solution representations.

Solution Construction via Spectral Decomposition

For the main mixed problem, separation of variables and expansion in eigenfunctions of the degenerate operator reduce the PDE to a countably infinite family of scalar fractional ODEs in time, one for each spatial eigenmode. Each mode satisfies a fractional Cauchy problem with data and frequency dictated by the spectral decomposition of the initial and source terms.

The solution to each modal equation leverages the Mittag-Leffler function $0<\alpha<1$1, serving as the fractional generalization of the exponential. Closed-form expressions for the solution incorporate convolution terms and series involving eigenvalues (scaled by the degeneracy parameter) and Mittag-Leffler kernels, parametrized by the fractional order and the degree of time degeneration.

Existence and Regularity Results

• Case $0<\alpha<1$2 (Classical Solutions): Under regularity assumptions on initial data and source in the appropriate weighted Sobolev spaces, the constructed solution converges absolutely and uniformly to a classical solution. The series expansions for $0<\alpha<1$3, and its relevant derivatives, are established to be valid via uniform boundedness of weighted sums of Fourier coefficients (with powers of eigenvalues), resorting to Bessel-type inequalities and Parseval's identities adapted to the degenerate context.
• Case $0<\alpha<1$4 (Weak Solutions): The degenerate operator is too singular for pointwise solutions—only weak solutions in space-time Sobolev spaces are meaningful. Here, orthogonality of the eigenfunction system in $0<\alpha<1$5 underpins the solution theory, and the main existence argument is via energy estimates and uniform convergence of mode sums. Regularity constraints are adjusted (data in $0<\alpha<1$6 and $0<\alpha<1$7 in time).

Uniqueness

Uniqueness is derived by considering the homogeneous version of the problem and showing that the only solution with vanishing data is the null function. The argument proceeds by spectral expansion: each modal coefficient solves a homogeneous time-fractional ODE with a unique solution (zero) due to the properties of the Caputo-like hyper-Bessel derivative, and completeness of the eigenbasis implies the triviality of the solution in the function space.

Implications and Future Directions

The analysis establishes a rigorous framework for PDEs that interleave time-fractional operators (with memory and possible time-degeneration) and spatial operators with degenerating coefficients, extending existing classical and fractional paradigms. The results have relevance for anomalous diffusion, heat conduction in inhomogeneous solids, and processes in biological or porous media with spatially variable properties.

Practically, the work provides a template for handling more complex spatial domains, higher-order time-fractional dynamics (multi-term or distributed order), and for numerical schemes leveraging spectral decomposition. Theoretically, the dependence of well-posedness and regularity on the degeneracy index opens further lines of inquiry into controllability, inverse problems, and stochastic analogs where noise interacts with degenerate and memory-laden dynamics.

Conclusion

This paper provides a comprehensive solvability analysis for mixed boundary value problems associated with time-fractional PDEs with time-space degenerating coefficients. It demonstrates, for broad parameter regimes, existence and uniqueness of solutions, clarifies the boundary condition landscape as affected by degeneration, and supplies fully explicit solution formulae based on spectral theory and Mittag-Leffler operators. The techniques and results furnish a theoretical platform for more advanced models of diffusion and conduction phenomena with nonlocal and inhomogeneous features.

Reference: "Solvability of a Mixed Problem for a Time-Fractional PDE with Time-Space Degenerating Coefficients" (2604.03739)