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Of current mathematical and physical interest are equatorial waves, in the setting of an azimuthal two-dimensional inviscid flow in a two-layer fluid with a flat bed and a free surface. The Hamiltonian formulation for nonlinear governing equations is adequate for structure-preserving perturbations, at the linear and at the nonlinear level have been derived. Since the ocean energy is concentrated in the long-wave propagation modes, it is of interest to pursue an in-depth analysis of long waves. In the present paper we will consider the process in the absence of vorticity.

For s 3 / 2 , it is known (see [24] ) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces H s when b > 1 . The proof of ill-posedness depends naturally on the value of b, and is based on the construction of peakon–antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for b 1 is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case b 1 on the line.

Background: Several studies indicate, particularly in the case of [18F]PSMA-1007, a relatively high rate of detection of ganglia in PSMA PET imaging. Ganglia are an integral part of the sympathetic portion of the autonomous nervous system. To date, no studies have directly compared [68Ga]Ga-PSMA-11 and [18F]PSMA-1007 ganglionic uptake intra-individually and analyzed the underlying molecular and physical mechanisms of different detection rates. With this monocentric retrospective study, we sought to evaluate the intra-individual physiological ganglion uptake of these different PSMA ligands in evidence-based imaging for prostate cancer. Methods: Our cohort consists of 19 male patients (median age 72 ± 9 with a range of 56–85) with biochemical recurrence of prostate cancer who underwent both [68Ga]Ga-PSMA-11 and [18F]PSMA-1007 PET/CT in our clinic on the same scanner per standard care between March 2015 and March 2022. Tracer uptake was quantified according to maximum standardized uptake value (SUVmax) for both [68Ga]Ga-PSMA-11 and [18F]PSMA-1007 PET/CT scans. Ganglia-to-background ratios (GBRs) were determined to quantify the image contrast through dividing the SUVmax of the ganglia by the background value (SUVmax of blood pool in the descending aorta, fatty tissue, and skeletal muscle in gluteal region). We used descriptive analyses for demographics and tumor characteristics and performed two-way repeated-measures ANOVA (analysis of variance) for SUV metrics including GBR measurements. Results: In total, we examined 101 ganglia with [18F]PSMA-1007 scanning, localized mostly in pairs as stellate, coeliac, and sacral, of which 76 were also detected with [68Ga]Ga-PSMA-11 PET/CT scanning. There was no statistically significant difference in PSMA uptake in terms of SUVmax between [18F]PSMA-1007 and [68Ga]Ga-PSMA-11 (p value: 0.052). In contrast, the comparison of GBRs revealed a higher detectability rate of ganglia with [18F]PSMA-1007 imaging (p < 0.001). Furthermore, a separate comparison of ganglia with respect to their anatomical location also demonstrated statistically significant differences both within and between [18F]PSMA-1007 and [68Ga]Ga-PSMA-11 PET/CT scans. Conclusion: Given the impression of more accentuated [18F]PSMA-1007 uptake in ganglia compared with 68Ga-labelled counterparts, our study demonstrated that the better detectability of ganglia is not due to more intense [18F]PSMA-1007 uptake by these small structures but to much more favorable physical properties of the radionuclide 18F. The most relevant limitations of our study are its retrospective design and the small patient cohort.

Background: Radiolabeled fibroblast activation protein (FAP) ligands, a novel class of tracers for PET/CT imaging, have demonstrated very promising results in various oncological, as well as in some benign, diseases with long-term potential to supplant the current pan-cancer agent [18F]FDG in some cancer types. Pancreatic ductal carcinoma (PDAC) belongs to the group of epithelial malignancies with a strong so-called “desmoplastic reaction”, leading to a prominent tumor stroma with cancer-associated fibroblasts that exhibit a marked overexpression of fibroblast activation protein (FAP). The first clinical experiences in PDAC with 68Ga-labeled FAP ligands suggested superior sensitivity to [18F]FDG. However, there is limited data with 18F-labeled FAP derivatives, i.e. [18F]FAPI-74, yet prospective single- and multicenter trials are already ongoing. In this proof-of-concept study, we sought to evaluate the biodistribution, tumor uptake, and lesion detectability in patients with PDAC using [18F]FAPI-74 PET/CT as compared to [18F]FDG PET/CT scans for staging. Methods: This study includes 7 patients (median age 69) who underwent both [18F]FDG PET/CT with contrast-enhancement and [18F]FAPI-74 PET with low-dose CT for primary staging (n = 3) and therapy response control after neoadjuvant (n = 1) or re-staging after palliative therapy (n = 3). The mean interval between PET scans was 11 ± 4 days (range 1–15 days). The [18F]FDG and [18F]FAPI-74 PET/CT scans were acquired at 64 ± 4.1 min (range 61–91 min) and 66.4 ± 6.3 min (range 60–76 min), respectively, after administration of 200 ± 94 MBq (range 79–318 MBq) and 235 ± 88 MBq (range 90–321 MBq), respectively. Quantification of tracer uptake was determined with SUVmax and SUVmean. Furthermore, the tumor-to-background ratio (TBR) was derived by dividing the SUVmax of tumor lesions by the SUVmax of adipose tissue, skeletal muscle, and blood pool. Results: Overall, 32 lesions were detected in 7 patients including primary (n = 7), lung (n = 7), bone (n = 3), lymph node (n = 13), and peritoneal metastases (n = 2). [18F]FAPI-74 detected 22% more lesions compared with [18F]FDG with a better TBR and visual lesion delineation. In one patient the primary lesion could be detected unequivocally with [18F]FAPI-74 but was missed by [18F]FDG imaging. Altogether, most of the lesions demonstrated markedly elevated uptake of [18F]FAPI-74 with a simultaneous lower uptake in the background, providing a very high visual contrast. Conclusion: To the best of our knowledge, this is the first, prospective, intra-individual investigation comparing [18F]FAPI-74 with [18F]FDG imaging in PDAC with encouraging results. These pivotalresults supporta larger, multicentric, prospective study to determine the value of [18F]FAPI-74 in detecting and staging PDAC in comparison with current standard of care imaging.

We consider the propagation of equatorial waves of small amplitude, in a flow with an underlying non-uniform current. Without making the too restrictive rigid-lid approximation, by exploiting the available Hamiltonian structure of the problem, we derive the dispersion relation for the propagation of coupled long-waves: a surface wave and an internal wave. Also, we investigate the above-mentioned model of wave-current interactions in the general case with arbitrary vorticities.

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

Simple Summary FAPI represents a novel class of radiotracers demonstrating promising results in terms of a high uptake in concordance with low background noise in several malignancies. Thereby, FAPI-PET/CT achieves sharp contrasts facilitating staging as well as tumor delineation and detection. However, FAP is also overexpressed for several non-oncological reasons allowing for benign indications as well. This review summarizes the current state of oncological and non-oncological FAPI-PET/CT in accordance with FAP in order to highlight future perspectives and identify areas where research is urgently warranted. Abstract A fibroblast activation protein (FAP) is an atypical type II transmembrane serine protease with both endopeptidase and post-proline dipeptidyl peptidase activity. FAP is overexpressed in cancer-associated fibroblasts (CAFs), which are found in most epithelial tumors. CAFs have been implicated in promoting tumor cell invasion, angiogenesis and growth and their presence correlates with a poor prognosis. However, FAP can generally be found during the remodeling of the extracellular matrix and therefore can be detected in wound healing and benign diseases. For instance, chronic inflammation, arthritis, fibrosis and ischemic heart tissue after a myocardial infarction are FAP-positive diseases. Therefore, quinoline-based FAP inhibitors (FAPIs) bind with a high affinity not only to tumors but also to a variety of benign pathologic processes. When these inhibitors are radiolabeled with positron emitting radioisotopes, they provide new diagnostic and prognostic tools as well as insights into the role of the microenvironment in a disease. In this respect, they deliver additional information beyond what is afforded by conventional FDG PET scans that typically report on glucose uptake. Thus, FAP ligands are considered to be highly promising novel tracers that offer a new diagnostic and theranostic potential in a variety of diseases.

This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, u t − u x x t + [ f ( u ) ] x − [ f ( u ) ] x x x + [ g ( u ) + f ″ ( u ) 2 u x 2 ] x = 0 that includes the Camassa–Holm equation as well as the hyperelastic-rod wave equation ( f ( u ) = k u 2 / 2 and g ( u ) = ( 3 − k ) u 2 / 2 ) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u 0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.

We study the Cauchy problem for the weakly dissipative Dullin–Gottwald–Holm equation describing unidirectional propagation of surface waves in a shallow water regime: u t − α 2 u x x t + c 0 u x + 3 u u x + γ u x x x + λ ( u − α 2 u x x ) = α 2 ( 2 u x u x x + u u x x x ) . In the present paper we demonstrate the simple conditions on the initial data that lead to blow-up of the solution in finite time.

The subject of this investigation is the long time behavior of the positive solutions of the following nonhomogeneous equation $$\begin{aligned} \rho \left( \left| x\right| \right) \frac{\partial \beta \left( v\right) }{\partial t}-\overset{N}{\underset{i=1}{\sum }}D_{i}(|D_{i}v|^{ \lambda -1}D_{i}v)+\rho \left( \left| x\right| \right) g\left( \beta \left( v\right) \right) +l\beta ^{1+m}\left( v\right) =f\left( x\right) \qquad \end{aligned}$$ (1) in unbounded domain \( \mathbb {R} _{+}\times \mathbb {R} ^{N},\) where the term \(g\left( s\right) \) is supposed to satisfy a condition \(g^{\prime }\left( s\right) >-l_{1}\) and \(D_{i}=\partial _{x_{i}}\) . The existence of the global attractor for the Eq. (1) in \(L^{1+\theta }\left( \mathbb {R} ^{N},\rho \right) =\left\{ v;v\rho ^{1/(1+\theta )}\in L^{1+\theta }\left( \mathbb {R} ^{N}\right) \right\} \) is proved.

We study the Cauchy problem for the weakly dissipative Camassa–Holm equation: $$u_{t}-u_{txx}+3uu_{x}+\lambda \left( u-u_{xx} \right)=2u_{x}u_{xx}+uu_{xxx}, \quad t >0 ,x\in \mathbb{R}$$ . A new blow-up result for positive strong solutions of the equation with certain profiles is presented. In particular, we use a condition where the initial data u 0 and its derivative u 0x are not simultaneously involved and the parameter λ is not bounded from above.

The subject of this investigation is existence and nonexistence of positive solutions of the following nonhomogeneous equation $ \rho (|x|) \frac{\partial u}{\partial t}- \sum_{i=1}^N D_i(u^{m-1}|D_i u|^{\lambda -1}D_i u)+g(u)+lu=f(x)$ (1) or, after the change $v=u^{\sigma}$, $\sigma =\frac{m+\lambda -1}{\lambda }, $ of equation $\rho (|x|) \frac{\partial v^{\frac{1}{ \sigma }}}{\partial t}-\sigma ^{-\lambda }\sum_{i=1} ^N D_i(|D_i v|^{\lambda -1}D_i v)+g(v^{\frac{1}{\sigma }}) +lv^{\frac{1}{ \sigma }}=f(x),$ (1') in unbounded domain $R_+\times R^N,$ where the term $g(s)$ is supposed to satisfy just a lower polynomial growth condition and $g'(s) > -l_1$. The existence of the solution in $ L^{1+1/\sigma}(0, T; L^{1+1/\sigma}(R^N))\cap L^{\lambda +1}(0, T; W^{1,\lambda +1}(R^N))$ is proved. Also, under some condition on $g(s)$ and $u_0$ is shown a nonexistence of the solution.

The subject of this investigation is the blow-up phenomena of the positive solutions of the mixed problem for the one-dimensional polytropic filtration equation with inhomogeneous density and source. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time. Note that the technique applied for the proof does not use the Zel'dovich-Kompaneets-Barenblatt solutions, since the construction of such type of function is more complicated in our case. Therefore, we obtain a result by multiplying on a special factor which has convenient properties. In particular, by choosing the parameters of the factor and using the properties of the solution, we obtain the inequality which allows us to show the blow up phenomena. (C) 2015 AIP Publishing LLC.

In this paper, we consider the dissipative Camassa-Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated. (C) 2014 Elsevier Inc. All rights reserved.

For a rapidly spatially oscillating nonlinearity 𝑔 we compare solutions 𝑢 𝜖 of non-Newtonian filtration equation 𝜕 𝑡 𝛽 ( 𝑢 𝜖 ) − 𝐷 ( | 𝐷 𝑢 𝜖 | 𝑝 − 2 𝐷 𝑢 𝜖 + 𝜑 ( 𝑢 𝜖 ) 𝐷 𝑢 𝜖 ) + 𝑔 ( 𝑥 , 𝑥 / 𝜖 , 𝑢 𝜖 ) = 𝑓 ( 𝑥 , 𝑥 / 𝜖 ) with solutions 𝑢 0 of the homogenized, spatially averaged equation 𝜕 𝑡 𝛽 ( 𝑢 0 ) − 𝐷 ( | 𝐷 𝑢 0 | 𝑝 − 2 𝐷 𝑢 0 + 𝜑 ( 𝑢 0 ) 𝐷 𝑢 0 ) + 𝑔 0 ( 𝑥 , 𝑢 0 ) = 𝑓 0 ( 𝑥 ) . Based on an 𝜀 -independent a priori estimate, we prove that | | 𝛽 ( 𝑢 𝜖 ) − 𝛽 ( 𝑢 0 ) | | 𝐿 1 ( Ω ) ≤ 𝐶 𝜖 𝑒 𝜌 𝑡 uniformly for all 𝑡 ≥ 0 . Besides, we give explicit estimate for the distance between the nonhomogenized 𝐴 𝜖 and the homogenized 𝐴 0 attractors in terms of the parameter 𝜖 ; precisely, we show fractional-order semicontinuity of the global attractors for 𝜖 ↘ 0 ∶ d i s t 𝐿 1 ( Ω ) ( 𝐴 𝜖 , 𝐴 0 ) ≤ 𝐶 𝜖 𝛾 .