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We consider initial boundary value problems for time fractional diffusion-wave equations: $$ d_t^{\alpha} u = -Au + \mu(t)f(x) $$ in a bounded domain where $\mu(t)f(x)$ describes a source and $\alpha \in (0,1) \cup (1,2)$, and $-A$ is a symmetric ellitpic operator with repect to the spatial variable $x$. We assume that $\mu(t) = 0$ for $t > T$:some time and choose $T_2>T_1>T$. We prove the uniqueness in simultaneously determining $f$ in $\Omega$, $\mu$ in $(0,T)$, and initial values of $u$ by data $u\vert_{\omega\times (T_1,T_2)}$, provided that the order $\alpha$ does not belong to a countably infinite set in $(0,1) \cup (1,2)$ which is characterized by $\mu$. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.

We consider an initial boundary value problem in a bounded domain $\Omega$ over a time interval $(0, T)$ for a time-fractional wave equation where the order of the fractional time derivative is between $1$ and $2$ and the spatial elliptic operator has time-independent coefficients and is not necessarily symmetric. We prove that if for arbitrarily chosen subdomain $\omega\subset \Omega$ and $T>0$, a solution to the problem vanishes in $\omega \times (0,T)$, then $u=0$ in $\Omega \times (0, T)$. The uniqueness does not require any geometric condition on $\omega$.

Our purpose is to adapt the Hilbert Uniqueness Method by J.-L. Lions in the case of fractional diffusion-wave equations. The main difficulty is to determine the right shape for the adjoint system, suitable for the procedure of HUM.

Viscoelastic materials have the properties both of elasticity and viscosity. In a previous work we investigate glass relaxation in the framework of viscoelasticity. Here we consider the Burgers model, a first but meaningful step in the general analysis, showing a reachability theorem thanks the analysis of the gap between eigenvalues and the representation of the solution in Fourier series.

Our goal is to establish a hidden regularity result for solutions of time fractional Petrovsky system where the Caputo fractional derivative is of order $\alpha\in(1,2)$. We achieve such result for a suitable class of weak solutions., Comment: arXiv admin note: text overlap with arXiv:2106.07416

We introduce a notion of weak solution for abstract fractional differential equations, motivated by the definition of Caputo derivative. We prove existence results for weak and strong solutions. We also give two examples as application of our results: time-fractional wave equations and time-fractional Petrovsky systems.

We will give some regularity results about fractional diffusion-wave equations.

We investigate the effect of memory terms on the entropy decay of the solutions to equations with Ornstein-Uhlenbeck operators. Our assumptions on the memory kernels include Caputo-Fabrizio operators and, more generally, the stretched exponential functions. We establish a sharp rate decay for the entropy. Examples and numerical simulations are also given to illustrate the results.

Here we show a hidden regularity result for nonlinear wave equations with an integral term of convolution type and Dirichlet boundary conditions. Under general assumptions on the nonlinear term and on the integral kernel we are able to state results about global existence of strong and mild solutions without any further smallness on the initial data. Then we define the trace of the normal derivative of the solution showing a regularity result. In such a way we extend to integrodifferential equations with nonlinear term well-known results available in the literature for linear wave equations with memory.

We take advantage of the approximation of the stretched exponential function with a general Prony series in glass relaxation to give some results about the spectral analysis for the equation of viscoelasticity. Moreover, in the case of the Burgers model we carry out a complete investigation that leads to the representation of the solution.

We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theory.

In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.

We consider an anisotropic hyperbolic equation with memory term: $$ \partial_t^2 u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_ju) + \int^t_0 \sum_{| \alpha| \le 2} b_{\alpha}(x,t,\eta)\partial_x^{\alpha}u(x,\eta) d\eta + F(x,t) $$ for $x \in \Omega$ and $t\in (0,T)$ or $\in (-T,T)$, which is a model equation for viscoelasticity. First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary $\Gamma \times (-T,T)$. Second we apply the Carleman estimate to establish a both-sided estimate of $| u(\cdot,0)|_{H^3(\Omega)}$ by $\partial_{\nu}u|_{\Gamma\times (0,T)}$ under the assumption that $\partial_tu(\cdot,0) = 0$ and $T>0$ is sufficiently large, $\Gamma \subset \partial\Omega$ satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in $F(x,t)$ and we establish a both-sided Lipschitz stability estimate.

We investigate control problems for wave-Petrovsky coupled systems in the presence of memory terms. By writing the solutions as Fourier series, we are able to prove Ingham type estimates, and hence reachability results. Our findings have applications in viscoelasticity theory and linear acoustic theory.