Semilinear Fractional Diffusion-Wave Models

• The semilinear fractional diffusion-wave equation replaces classical time derivatives with a Caputo derivative of order between 1 and 2, combining linear spatial operators with nonlinear reaction terms to interpolate between diffusive and wave regimes.
• It includes multiple formulations, employing spectral decompositions, mild solution theories, and critical Lebesgue-space analyses to address bounded-domain and whole-space problems.
• Advanced numerical methods, such as the SFOR approach and adaptive discretizations, are developed to handle initial-layer singularities and multi-term memory effects in the equation.

Searching arXiv for recent and foundational papers on semilinear fractional diffusion–wave equations and closely related inverse/numerical theory. A semilinear fractional diffusion-wave equation is an evolution equation in which the classical first or second time derivative is replaced by a Caputo derivative of order $1<\alpha<2$, and the linear spatial part is combined with a nonlinear reaction term. In its prototypical bounded-domain form, it reads

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$

or, in Laplacian form,

$\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$

supplemented with Dirichlet boundary conditions and the two initial conditions $u(0,x)=u_0(x)$, $u_t(0,x)=u_1(x)$. The regime $1<\alpha<2$ interpolates between diffusion at $\alpha=1$ and the classical wave equation at $\alpha=2$, and the resulting dynamics combine memory effects with weaker smoothing than in the parabolic case (Kian et al., 2015, Andrade et al., 6 Sep 2025).

1. Canonical formulations and scope

The standard semilinear fractional diffusion-wave problem on a bounded domain $\Omega\subset\mathbb R^d$ is posed as

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$

with homogeneous Dirichlet boundary condition on $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$0, and

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$1

where $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$2, $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$3 is the Caputo derivative, and $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$4 is either the Dirichlet realization of a second-order elliptic operator or the Dirichlet Laplacian (Kian et al., 2015, Costa et al., 6 Feb 2026). A widely used structural assumption is that the nonlinearity is $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$5, vanishes at zero, and has polynomial-type derivative growth,

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$6

so that $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$7 is the model example (Kian et al., 2015).

A parallel $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$8-based formulation replaces $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$9 by a locally Lipschitz power-type nonlinearity $\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$0 satisfying

$\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$1

and studies mild solutions with initial data $\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$2 by means of fractional power scales associated with the Dirichlet Laplacian (Andrade et al., 6 Sep 2025, Costa et al., 6 Feb 2026).

A closely related but distinct class consists of semilinear fractional diffusion equations with a spatial fractional Laplacian and classical first-order time derivative,

$\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$3

This is a fractional diffusion equation, not a diffusion-wave equation in the time-fractional sense; however, it is explicitly presented as a prototype for more general fractional diffusion-wave models with $\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$4 (Li, 2021).

Formulation	Structural feature	Source
$\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$5	$\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$6, elliptic $\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$7, Dirichlet data	(Kian et al., 2015)
$\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$8	$\partial_t^\alpha u(t,x)=\Delta u(t,x)+f(u(t,x)),$9-based mild theory, power-type $u(0,x)=u_0(x)$0	(Andrade et al., 6 Sep 2025, Costa et al., 6 Feb 2026)
$u(0,x)=u_0(x)$1	space-fractional diffusion, not time-fractional diffusion-wave	(Li, 2021)

This terminological boundary matters. In the current literature, “fractional diffusion-wave” refers primarily to the order of the time derivative, whereas “fractional diffusion” may instead refer to a nonlocal spatial operator.

2. Linear operator families, spectral structure, and mild solutions

The linear backbone of the theory is the fractional evolution problem

$u(0,x)=u_0(x)$2

with $u(0,x)=u_0(x)$3 the positive selfadjoint Dirichlet realization of an elliptic operator. Spectral decomposition with respect to the eigenpairs $u(0,x)=u_0(x)$4 of $u(0,x)=u_0(x)$5 reduces the problem to scalar fractional ODEs

$u(0,x)=u_0(x)$6

whose solutions are expressed באמצעות Mittag–Leffler functions. In Hilbert-space form, the solution admits the representation

$u(0,x)=u_0(x)$7

where $u(0,x)=u_0(x)$8 are operator families built from $u(0,x)=u_0(x)$9, $u_t(0,x)=u_1(x)$0, and $u_t(0,x)=u_1(x)$1 (Kian et al., 2015).

In the $u_t(0,x)=u_1(x)$2-theory, the same role is played by the resolvent families

$u_t(0,x)=u_1(x)$3

defined through inverse Laplace transforms of $u_t(0,x)=u_1(x)$4. The corresponding mild solution formula is

$u_t(0,x)=u_1(x)$5

These families satisfy smoothing estimates of the form

$u_t(0,x)=u_1(x)$6

with analogous bounds for $u_t(0,x)=u_1(x)$7 and $u_t(0,x)=u_1(x)$8, which are central to both existence and regularity arguments (Andrade et al., 6 Sep 2025).

A complementary PDE realization arises in space-time fractional wave equations

$u_t(0,x)=u_1(x)$9

where the spectral fractional power $1<\alpha<2$0 is represented as a Dirichlet-to-Neumann map for a nonuniformly elliptic extension problem on the semi-infinite cylinder $1<\alpha<2$1. In that framework, the trace $1<\alpha<2$2 solves the original equation, while the extension $1<\alpha<2$3 satisfies a weighted elliptic equation in the extended variable and a dynamic boundary condition at $1<\alpha<2$4 (Otarola et al., 2017).

For the one-dimensional whole-space linear equation

$1<\alpha<2$5

the fundamental solution is explicit in terms of Mittag–Leffler and Wright–Mainardi functions. Although the support is noncompact for $1<\alpha<2$6, the maximum of the Green function occurs at

$1<\alpha<2$7

and the maximum value satisfies

$1<\alpha<2$8

so the disturbance propagates infinitely fast while the maximum disperses with finite speed (Luchko et al., 2012). This linear picture underlies mild-solution heuristics, kernel estimates, and critical-exponent analysis in the semilinear theory.

3. Local well-posedness and regularity regimes

One local existence theory is based on Strichartz-type estimates for the linear fractional wave equation. For $1<\alpha<2$9 or $\alpha=1$0, $\alpha=1$1, and nonlinearities $\alpha=1$2 with $\alpha=1$3 and $\alpha=1$4, local weak solutions exist provided

$\alpha=1$5

With suitable parameters $\alpha=1$6, the solution belongs to

$\alpha=1$7

and the lifespan can be chosen explicitly as

$\alpha=1$8

This yields longer existence times for smaller initial data (Kian et al., 2015).

A second framework develops local mild solutions directly in Lebesgue spaces. Let $\alpha=1$9, and suppose

$\alpha=2$0

together with $\alpha=2$1. Then for initial data in a ball of $\alpha=2$2 there exists a unique mild solution

$\alpha=2$3

which instantaneously gains spatial regularity: $\alpha=2$4 and satisfies the weighted estimate

$\alpha=2$5

for $\alpha=2$6. The same theory provides a continuation principle and the blow-up alternative

$\alpha=2$7

This is the natural $\alpha=2$8-analogue of parabolic semilinear theory, but with resolvent families replacing analytic semigroups (Andrade et al., 6 Sep 2025).

A third regime is the critical Lebesgue-space theory. For

$\alpha=2$9

the problem is critical in the sense of local well-posedness of mild solutions in $\Omega\subset\mathbb R^d$0. In this setting one introduces $\Omega\subset\mathbb R^d$1-regular mild solutions satisfying

$\Omega\subset\mathbb R^d$2

under the condition $\Omega\subset\mathbb R^d$3. Local well-posedness, Lipschitz dependence on initial data, uniqueness within the mild class, and a maximal continuation interval are all proved in this critical framework (Costa et al., 6 Feb 2026).

Framework	Core assumptions	Local solution class
Strichartz-type Hilbert theory	$\Omega\subset\mathbb R^d$4, $\Omega\subset\mathbb R^d$5	$\Omega\subset\mathbb R^d$6
Lebesgue mild theory	$\Omega\subset\mathbb R^d$7, $\Omega\subset\mathbb R^d$8	$\Omega\subset\mathbb R^d$9 with smoothing into $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$0
Critical Lebesgue theory	$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$1, $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$2	$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$3-regular mild solutions

These theories are complementary rather than redundant. The first is tuned to Strichartz admissibility and Sobolev embeddings; the second and third are formulated directly in fractional power scales associated with the Laplacian and are especially effective for low-regularity $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$4 data.

4. Critical exponents, global existence, and blow-up structure

For the whole-space equation

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$5

there are two distinct critical exponents for global small-data solutions. If both initial data $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$6 and $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$7 are generically nonzero, the critical exponent is

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$8

If the second datum vanishes, $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),\qquad (t,x)\in (0,T)\times\Omega,$$9, the critical exponent improves to

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$00

The second threshold reflects a genuinely fractional effect: the memory structure changes the decay properties of the linear response, so the critical power is not determined by naive scaling alone (D'Abbicco et al., 2017).

In the same whole-space setting, global solutions with small initial data in $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$01 exist for $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$02 above the relevant threshold, and the decay rate depends on whether $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$03 vanishes. With both data present, the solution decays like

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$04

whereas in the case $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$05 the decay becomes

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$06

The appearance of two different exponents is one of the distinctive features of semilinear fractional diffusive equations of order $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$07 (D'Abbicco et al., 2017).

In bounded domains, the global theory is more selective. The local theory of semilinear fractional wave equations in the Strichartz framework does not generally yield global existence, and the lack of a convenient notion of conserved or monotone energy adapted to the Caputo derivative is identified as the main obstacle (Kian et al., 2015). By contrast, the critical $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$08-theory proves global $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$09-regular mild solutions for sufficiently small data $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$10, $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$11, with uniform bound

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$12

It also proves an asymptotic equivalence theorem: for two global small-data solutions $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$13,

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$14

if and only if the same holds for the difference of the corresponding linear evolutions. In this regime, the nonlinearity does not change the leading-order asymptotics (Costa et al., 6 Feb 2026).

These results suggest a stratified global picture. Whole-space power nonlinearities admit a sharp small-data threshold theory with two critical exponents; bounded-domain critical $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$15 problems admit global small-data solutions and linear asymptotics; and more general bounded-domain semilinear problems remain largely local because the Caputo-wave setting lacks the standard conserved structures of classical semilinear wave equations.

5. Spatially nonlocal variants and inverse formulations

A substantial extension of the subject replaces the local spatial operator by a fractional power. In the linear space-time fractional wave equation

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$16

the operator $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$17 is realized as a Dirichlet-to-Neumann map for a nonuniformly elliptic problem on $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$18. The trace $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$19 of the extension solves the original equation, while $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$20 satisfies a weighted elliptic equation in the extended variable $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$21 and a dynamic boundary condition involving $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$22 on $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$23. This extension framework yields existence, uniqueness, energy estimates, and anisotropic space-time regularity, and it is explicitly identified as a foundation for semilinear problems of the form

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$24

(Otarola et al., 2017).

A different spatially nonlocal direction concerns inverse problems. The semilinear equation

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$25

with fractional-power nonlinearity

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$26

is not a diffusion-wave equation because its time derivative is classical, but it supplies a prototype for nonlocal semilinear inverse theory. For small exterior Dirichlet data, one defines a nonlocal Dirichlet-to-Neumann map

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$27

and partial exterior measurements on two open subsets $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$28 determine all coefficients $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$29. The proof uses only first-order linearization and a parabolic Runge approximation property, rather than higher-order linearization (Li, 2021).

The broader spatially fractional picture also includes the wave extension problem

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$30

for which the generalized normal derivative at $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$31 recovers $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$32. This produces oscillatory subordination formulas, Bessel-function representations, and explicit kernels that the paper presents as a linear foundation for semilinear equations involving fractional spatial operators (Kemppainen et al., 2014).

A common misconception is that all “fractional diffusion-wave” equations are fractional only in time. The current literature distinguishes at least three analytically different situations: time-fractional semilinear diffusion-wave equations, space-fractional semilinear diffusion equations, and genuinely space-time fractional wave equations. Their techniques overlap, but the operator-theoretic and inverse-problem structures are not identical.

6. Numerical analysis, order reduction, and regularity constraints

The most specific semilinear numerical framework for $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$33 is based on the symmetric fractional-order reduction (SFOR) method. Starting from

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$34

with $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$35, one sets $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$36, subtracts the singular linear term

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$37

and defines

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$38

Then

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$39

so the original equation is rewritten as a coupled system in which both time-fractional operators have order $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$40. This symmetry permits the use of the same nonuniform L1 or Alikhanov discretization in both equations, together with a linearly implicit treatment of the nonlinearity (Lyu et al., 2021).

On general nonuniform meshes, the resulting schemes satisfy discrete kernel monotonicity and lower-bound properties that support an $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$41-energy method. Under mesh assumption MA and regularity assumptions on $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$42 and the auxiliary variable $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$43, the nonuniform L1 scheme achieves temporal accuracy up to order

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$44

while the nonuniform Alikhanov scheme achieves temporal accuracy up to order $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$45; both retain spatial accuracy $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$46 in the discrete $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$47-type norm used in the analysis (Lyu et al., 2021). The same paper designs an adaptive time-stepping strategy based on the difference between fast L1 and fast Alikhanov approximations.

At the same time, the regularity assumptions underlying time discretization must be treated with care. For linear space-time fractional wave equations, detailed PDE analysis shows that the usual assumptions often made in numerical analysis are problematic: even for smooth data, the solution behaves near $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$48 like

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$49

so that

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$50

and one only obtains weighted estimates such as

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$51

Thus high-order time smoothness at $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$52 is structurally incompatible with the genuine fractional diffusion-wave dynamics (Otarola et al., 2017).

This point is not merely technical. It explains why graded meshes, nonuniform time stepping, and singularity-aware order reduction are recurrent in the numerical literature on fractional diffusion-wave equations, and why the best-performing semilinear schemes are built around the precise initial-layer structure rather than around classical $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$53-in-time assumptions.

7. Multi-term memory kernels and adjacent generalizations

The single-term Caputo operator is only one instance of a broader family of anomalous time evolutions. Linear generalized diffusion-wave equations replace the second time derivative by an integro-differential operator with memory kernel,

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$54

which includes standard time-fractional and distributed-order diffusion-wave equations as special cases. In Laplace–Fourier variables, the fundamental solution has the universal form

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$55

and the paper derives Green functions, non-negativity conditions, and mean squared displacement laws for power-law, distributed-order, tempered, and Prabhakar kernels (Sandev et al., 2019). Although this theory is linear, it is explicitly presented as the catalogue of kernels and operators one uses when nonlinearities are added.

A semilinear analogue in the subdiffusive regime studies

$$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$56

with multi-term Caputo operators of orders $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$57. Under structural conditions on the coefficients and on $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$58, one obtains global $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$59-bounds, bounds on the memory term $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$60, and, for time-independent coefficients, absorbing sets in $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$61 or $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$62. In the constant-coefficient case without memory, decay is governed by multinomial Mittag–Leffler functions and asymptotically by the smallest fractional order (Vasylyeva, 2024).

These nearby generalizations clarify the landscape around semilinear fractional diffusion-wave equations. The single-order Caputo diffusion-wave model remains the canonical setting for $$\partial_t^\alpha u(t,x)+\mathcal A u(t,x)=f_b(u(t,x)),$$63, but multi-term operators, distributed-order kernels, and spatially fractional realizations show that the subject is better understood as part of a larger theory of semilinear evolution equations with memory, resolvent families, and nonlocal spatial generators.